Linner Statistics

Chi-square tests

2011-03-25T13:24:00.001-04:00

We're nearing the end of the new material. How sad!

HW for tonight: 14.3, 14.4, 14.5, and 14.8

Second semester 2011--Starting with chapter 9&10

2011-02-09T17:18:00.000-05:00

Welcome back, students!

AP REGISTRATION LINK: https://user.totalregistration.net/AP/111983
PLEASE LET YOUR TEACHER KNOW IF FINANCIAL HARDSHIP PREVENTS YOU FROM REGISTERING. WE'LL TRY TO FIND A WAY TO SOLVE THE ISSUE.

We begin by investigating the distributions of p-hat and x-bar. This is the concept of sampling distributions. We consider the distribution of ALL the sample means that we would observe if we took EVERY sample of size n from a population.

In class on Wednesday we collected data: we computed average penny ages from samples of sizes 5, 10, and 25. We have more collecting and computing to do before the distributions become evident from the graphs. Prepare to crank through more pennies on Thursday. (The program we used to report the findings is Fathom.)

Homework January 4th: Work one problem from the handout completely. Become a master of that problem.

Homework January 5th: Read pages 563-568 from the text and work problems 9.1, 9.2, and 9.5.

Be prepared for a quiz at any time.

Homework January 7th: Problem 9.7 using Excel if possible, problems 9.10 through 9.17.
_____________________________________________________________________________________

Homework January 20: Problems 9.31, 9.32, 9.33 HAVE THIS DONE BY MONDAY. Remember, your book should be read by Wednesday.

______________________________________________________________________________

February 9, 2011
You've been busy in class collecting data and constructing confidence intervals for the mean and for the proportion.
There are three cases to consider on tomorrow's test:

Confidence intervals for proportions
Check n phat and n(1-phat) and that n < 1/0 N
Use the sample proportion (phat) in your calculation of the standard error
Use a Z statistic for computing the margin of error
Don't forget the interpretation

Confidence intervals for the mean (when we miraculously KNOW the population standard deviation)
Check that the observed values would not indicate that the means would be non-normal
Are the observations random and independent?
Use the CLT--standard error = pop std dev / sqrt sample size
Use Z (remember, this is the miraculous case)
Don't forget the interpretation

Confidence intervals for the mean (when we miraculously DON'T KNOW the population standard deviation)
Check that the observed values would not indicate that the means would be non-normal
Are the observations random and independent?
Use the CLT--standard error = SAMPLE std dev / sqrt sample size
Use t-distribution with n-1 degrees of freedom
Don't forget the interpretation

Margin of error = (Z or t)* Std error
Greater confidence = wider margin of error
Larger sample size = smaller margin of error

Good interpretation of the confidence interval:
We are 95% confident that the true population mean test score falls between 3.2 and 3.6.

Good interpretation of the confidence level:
If this procedure were repeated many times, we would expect approximately 95% of the confidence intervals constructed from the sample mean test scores to contain the true population mean test score.

Bad interpretations:
Like you really expected me to post BAD examples? Anything that says there is a 95% chance. . . is really bad.

A confidence interval without an interpretation is relatively worthless. Almost as bad as using a point estimate instead of a confidence interval! Don't fall into the lazy trap of answering questions without including all the required parts.

Preparing for the 1st semester final

2010-12-13T09:48:00.001-05:00

Topics on the AP Stat 1st Semester Final Exam

Types of graphs, their advantages and disadvantages, their interpretations
Measures of center and spread, their calculation, their different meanings, and their uses
Measures of position, converting back and forth among different measures (i.e. percentile, observed value, and z-value)
Probabilities associated with continuous random variables (area under the curve, normalcdf, empirical rule, Chebyshev’s theorem)
Probabilities associated with discrete random variables, multiplication property, addition property, independence, conditional probabilities
Special discrete random variables—binomial and geometric distributions
Relationships in two variables – linear regression, residuals, interpreting regression output, correlation coefficient, coefficient of determination
Means and standard deviations of combinations and transformations of random variables
Design of surveys, types of bias, types of sampling
Design of experiments, methods of randomizing, methods of control, matched pairs design, blocking, causation
Vocabulary: for instance, outliers, clusters, gaps, population, sample, variance, influential observations

Answer the following two questions on two separate sheets of paper. Your response to question one will be graded as a small test grade. Your response to question two will be the free response part (take-home portion) of your final exam. Your answers may be hand-written or typed, but must be legible and complete. Computed numbers that are unsupported by their calculations will be given no credit. You may NOT work together on this assignment.

Question One: Using an example from the second half of your selected book (Bringing Down the House, Freakonomics, etc.), explain a specific connection to one of the topics in the list of exam topics above. You may get creative with your product for this question. It may be in the form of a Powerpoint, a 9”x12” poster, or other appropriate written or mixed media form. Interpretive dance is not appropriate.

Question Two: Answer the problem handed out in class on a separate sheet of paper. You must work alone on this problem.

Distributions of Random Variables

2010-12-06T17:10:00.001-05:00

11/19/2010
We're combining parts of chapters 6-8 to build on students' prior understanding of probability.

First up: Geometric and binomial probabilities
geometric probabilities:
Know the 4 characteristics that define a geometric distribution
Know how to find the expected value of x
Know how to find probabilities for values of x (both individual probabilities and cumulative probabilities)

bimomial probabilities:
geometric probabilities:
Know the 4 characteristics that define a binomial distribution
Know how to find the expected value of x
Know how to find probabilities for values of x (both individual probabilities and cumulative probabilities)

Be able to identify a binomial or geometric distribution when you read a problem.
Define the random variable x.
Solve problems related to probabilities for these distributions.

Next up: any other discrete distributions
Work with valid probability distributions (individual and cumulative)

Apply the concept of independent events to joint probability problems.

Apply the concepts of disjoint sets and complements to find probabilities.

Find the means and standard deviations of transformations of a random variable and combinations of independent random variables. YOu'll have to bookmark these pages and study them A LOT!

Monday December 6, 2010
Testing on Thursday on distributions of random variables. We will be learning new material through Wednesday. Be here!

Producing Data

2010-11-04T15:39:00.002-04:00

So we've moved into survey design and experimental design. (Chapter 5 in the text.)

Important vocabulary:
Population
sample
census
bias
nonreaponse bias
undercoverage
response bias
convenience sample
stratified random sample
systematic random sample (like The Lottery by Shirley Jackson)
simple random sample (SRS)

We used the Table of Random Digits to (1) pick a sample, (2) Simulate a random event, and (3) randomly allocate participants to experimental treatments.

We have looked at a few experimental design/survey design questions from previous exams.

HW due Thursday 11/4: Problems 5.2, 5.3, 5.10 ,5.11
HW due Friday 11/5: Written answers on your own paper to the 1998 and 2002 experimental design questions handed out in class.

Bivariate distributions

2010-10-22T12:27:00.000-04:00

We've moved from investigations using single variables to the world of two variables. The first type of bivariate relationship we study is the relationship between two numerical variables.

We collected data on Monday and crunched numbers again on Tuesday to find the least squares regression line and the coefficient of determination. R^2.

On Wednesday we studied the correlation coefficient, the slope and intercept of the LSRL, and the patterns in residuals. We saw that the point (x-bar, y-bar) lies on the LSRL and that R^2 may not be an indicator of a good model.

Based on your new knowledge of these concepts, please expand your 4 inch summary of section 3.1 to 5 inches of strong content.

Do the problems in section 3.1 that relate to the manatees and to the archaeopteryx.

Quiz answers:
scatterplot - points have a strong positive linear pattern with no outliers. Graph should have labels and scale.
LSRL y-hat = -10.64 + 4.117x
Residuals - y - y-hat graphed against x. To compute residuals, use L2 - Y1(L1). The graph shows that the residuals fluctuate above and below the axis with varying distances.
Interpretation - Because the residuals can be interpreted as randomly scattered about the residual = 0 line, the linear model is good.
Caveat- Because the residuals seem to be getting further from residuals = 0 as x gets larger, we might be concerned about our error increasing as length increases. Beware telescoping residuals.

October 8, 2010

This week student should have completed problems 3.35, 3.36, 3.37, & 3.37 from the text. For HW due Monday, they need to complete problems 3.39, 3.40, & 3.48.

What have we done so far? Collected bivariate data. Looked at them. Computed the LSRL. Computed residuals. Interpreted the residuals and the slope of the LSRL. Used the LSRL to predict a value of y. Performed a Linear regression t-test to determine the significance of the slope.

What do we have to do? Practice and interpret outputs.

ALSO, pick a book. Suggestions: A Civil Action, Freakonomics, Bringing Down the House, Moneyball, And the Band Played On, The Lady Tasting Tea. Get your parents' permission to read your book. You should have it finished by the end of Thanksgiving break.

Monday, October 11, 2010

We collected data that we expected to have no correlation. In 13 of the 15 cases, we got what we expected. We graphed the ordered pairs, computed the LSRL, checked the residuals, performed a linear regression t-test, and interpreted the results.
Small p-value>>> reject the null hypothesis--that there is no linear relationship between x and y. Instead, we have evidence indicating that there is a linear relationship.
Large p-value>>> fail to reject the null hypothesis. We do not have compelling evidence that there is a linear relationship.

HW problems 3.6 and 3.61

Have your papeback on Friday. You will be given a reading day.
_________________
As we continue through bivariate distributions, please take care to clearly identify the transformation you have performed on your lists in the calculator. For instance, log L2 may make sense to you, but you may be better off by renaming the list log life expectancy.

Typically, students have problems when they graph the curves through data. The linear regression graph only works with straightened (transformed) data. The curves go through the original data.

Your test on bivariate data will be Thursday, Oct 28.

Measurements of position

2010-08-30T13:54:00.003-04:00

Hmmm. Z values. Percentile ranks. Proportions between two x-values.
How are these connected for Normal distributions?

The percentile for a particular z-value is the value in the body of the Z table that represents the "sum" of the column and row titles. For Negative z-values, just append (attach) the hundredths place digit. For instance. . .
row 1.3, column 0.4 ==> 1.34 = z. This is the 90.99th percentile.
for row -2.3, column 0.4 ==> -2.34 = z. With a table value if 0.0096, this is just a hair under the 1st percentile.

The percentile is the proportion of data that lies to the left of the x value or is equal to it. If you took a test and scored at the 99th percentile, 99% of all other test scores should be equal to your score or below it.

Another way to find the percentile is to use the NormalCDF function on the calculator. Use NormalCDF(lower bound, upper bound) where the boundary values are z scores. To find the percentile for a Normally-distributed z value, we use the lower bound of negative infinity and the upper bound of the z under consideration.

We can use -999999 for negative infinity. NormalCDF(-999999,1) = the proportion of the population of Normally distributed z values that fall equal to or below 1.

To find the Z value for a particular percentile, use the inverse of the NormalCDF function-- INVNorm. To find the 95th percentile, enter InvNorm(.95). Approximately 95% of all z-values in a Normal distribution will fall below this value.

To find the X value that corresponds to the desired Z value, take the mean and add Z standard deviations.

Practice converting X values into z values adn percentiles into X values. Do the problems on page 147.

New data!!!! Haircut costs

2010-08-20T15:00:00.000-04:00

As we learn to represent and interpret our data, we collected the following data:
boys' haircut prices
12, 18, 22, 0, 0 ,0, 15, 0, 17, 16, 17.95, 10, 12
girls' haircut prices
35, 55, 50, 30, 18, 25, 0, 50, 40, 45, 45, 140, 40, 8, 25, 30, 22, 28

Represent each of these as a boxplot on the same axes AND
using the information starting on page 42 in the text, represent it also as a back to back stemplot.

We will interpret your results on Tuesday.

Be safe.
_________________________

8/20
We've used histograms, boxplots, and stemplots to represent univariate (one-dimensional) data. We've worked many problems from previous AP exams.

You're probably ready to close out this chapter (1). Let's focus on the parts we haven't covered so far and test on Thursday, 8/26.

We will start the CiCi's Sundays on August 29, unless you do not need help yet.

Be safe. Play hard. Go Trojans.

Gearing up for the final exam

2010-05-10T15:03:00.003-04:00

So far we've discussed good answers for problems 1-4 of the operational exam. The take-home part of your final will be one of the 6 questions, randomly selected two days before your exam. Get to work developing your best answers to these questions.

The in-class part of the final exam will be cumulative, with an emphasis on second semester topics.

Random assignment of problems for take-home portion:

6th period seniors: #3 YOU MAY NOT COLLABORATE ON THIS PROBLEM NOW THAT IT HAS BEEN ASSIGNED. THE WORK YOU TURN IN MUST BE YOUR OWN.

Chi-square procedures

2010-03-30T15:51:00.005-04:00

http://lassiterstatistics.wikispaces.com/

Send your summary documents (in pdf format if possible) to jhl2881 at
students dot kennesaw dot edu

Suggested problems from Chapter 14:
goodness of fit: 3, 4, 5 (this is an example of how biologists use 2x2 tables to do goodness of fit tests), 9 (simulation)
2-way tables: 13, 19, 20, 14, 16, 18, 12, 17, 22, 24, 32, 33, 34.

We're beginning our last new topic (since we already did linear regression inference once).

We will use two different types of chi-square procedures and three different names for the procedures.

First, if some higher power determined what the proportions of the sample should have been associated with different values of the categorical variable. . .
like what portion of your M&Ms should have been red, brown, green, etc., then you will use a Goodness of Fit test to compare your experience (the sample) with what the higher powers suggested. This is also the test we use when the higher power might suggest that the distribution should have been "fair" or equal across all the values.

If the sample itself is going to suggest a distribution, then we use the test on independence or the test of homogeneity. These two tests are performed the same way, we just have different inputs and hypotheses associated with the two forms.

When we have one sample from one population and want to know if characteristics are associated, like red hair and green eyes, we might use the test of indepenence.

If we have two populations, like smokers and non-smokers, and want to know if the two populations had the same propensity for speeding tickets, we could use a test of homogeneity with cleverly-selected data.

Methods will be discussed in class.

Have the printed draft of your assignment in class on Wednesday!

T-tests for means

2010-03-26T15:00:00.002-04:00

http://www.nytimes.com/2010/02/28/weekinreview/28sussman.html?ref=weekinreview
.
Time magazine article on the complications of the race and ethnicity entries on the census: http://www.time.com/time/nation/article/0,8599,1975883,00.html?hpt=T2
What inference procedure do I perform? applet (http://www.ltcconline.net/greenl/java/Statistics/StatsMatch/StatsMatch.htm?)

Welcome to the new stuff. Same as the old stuff. . .almost.

EQ: Why do you use inference?
Under what conditions can we use inference?

As you've seen, t-tests for means are quite similar to the z-tests for proportions. We still follow the same pattern of Setup-Check Assumptions-calculations/arithmetic-decision in the context of the problem.

Now, to use t-methods we have to prove that the distribution of x-bars is probably mound-shaped and symmetric enough to invoke the CLT. If given the data, sketch the histogram of the observations or the Normal probabilty plot. If either sample observation graph looks severely non-normal (with gaps or outliers), then we cannot assume that the means of the samples drawn from that type of population would be close to Normal.

Our quiz on 3/2 will be a lot like the lab we did in class on Monday. You will have data to analyze (SCAD).

3/4/10 Now you've taken two quizzes and shown great improvement.

Things you can do to improve your communication:

Label the graph of the observations or the Normal Probablity plot. The x axis of the histogram should be named with the thing you're measuring, like blood pressure. The scale should be added at about 5 places (no need to mark off every little bit). The y axis is the frequency. Help the reader out by labeling the tallest bar.

Label your Normal Probability plot with the definition of the x at the bottom. No scales are necessary if you label the NPP as a Normal Probability Plot.

Sketch the Normal curve before you start the calculations. Put the hypothesized mean in the middle. Then mark your x-bar to the right or the left (as appropriate). This will help you put the less than or greater than sign into your calculations correctly. It also reminds you that a probability cannot be less than 0 or more than 1!

Refine your decisions. Compare, conclude, contextualize, convince.

HW due Friday: Finish the AP exam problem begun in class that deals with a paired t-test. A paired t-test is performed exactly the same way as a regular -test, but on the third set of data--the difference between the two sets of DEPENDENT data.

PAIRED T-TESTS
This is used when the two "samples" are not independent, but two measures from each of the experimental units or participants. Examples: pre-tests and post-tests on the same students, the e.coli problem, the pharmacy problem, and the hand span problem.

Align the samples so you can subtract the "pre-test" value from the "post-test" value or perform a similar subtraction to generate one list of differences. Use this list as your input for the one-sample t-test.

T-TESTS FOR TWO INDEPENDENT SAMPLES
For this, Ho is mu1 - m2 = some number.

The test statistic is (x-bar 1 - x-bar 2)/std error of the difference of the means.

Use tcdf to find the area in the tail. The procedure is much like the regular t-test.

CONFIDENCE INTERVALS: estimate +- t* (std error of the estimate).

Use the x-bar or the difference of the x-bars as the estimate.
Use the sx/sqrt n or sqrt((Sx1)^2/n + (Sx2)^2 /n ) for the std error.

Work problems from the chapters for homework.

TEST REWORK/TEST "RETAKE"

There will be a test "retake" available on Thursday, March 18 in class for anyone who turns in their completed reworked problems from the Two-proportion test.

The retake will include one-proportion and two proportion tests and intervals, Type I and Type II error, and finding the sufficient sample size for a margin of error.

EXAM REVIEW
Have the draft of your first review page (assigned individually in class) with you in class on March 31. We will peer-review and edit the pages before they go into the class webpage. Go to this page to see examples of how the class of 2009 handled a similar assignment. Your product may be a webpage or a document.

HW for every night: Work problems from chapters 11-13. You should be able to do all these problems.

Essential questions that you need to write responses for:

Why is a t-test used instead of a z-test when we do not have the population standard deviation?

Why is it inappropriate to perform repeated tests instead of relying on one test?

The mean of x for sample 1 - the mean of x for sample 2 = the mean of the paired differences. So why does it matter whether we perform a two-sample test of independent means or a paired t-test? (This is incredibly important!!!)

WRITE AT LEAST A PAGE ABOUT THESE TOPICS. Hint: Consider the effect that your choice of test has on the power of the test.

TEST on inference for means: March 30, 2010

Inference for Proportions

2010-02-19T18:47:00.005-05:00

Congratulations, AP STAT GROUP! 100% participation in the AP exam!

Standards: IV A3, A4, A5, B3, B4, and now B1.
INFERENCES FOR ONE PROPORTION and the DIFFERENCE BETWEEN TWO PROPORTIONS.

We're combining aspects of Chapters 10, 12, and 13 in the text to understand inferences about proportions.

Please print out a copy of the complete hypothesis test and confidence interval examples from my Lassiter blog (http://lhsblogs.typepad.com/linner)

Some of the basics:
Every complete inference problem will have four parts: setup, assumptions, calculations, and decision in the context of the problem.

The set-up of a one-proportion z test will include the definition of the parameter of interest, the hypotheses, and any other information you will use to perform the test.

The assumptions portion includes checking all assumptions and conditions necessary to use the z test, in other words that the data are randomly selected, independent, from a Normally-distributed population, and allow us to use the simple standard deviation formula.

The calculations include the name or formula for the test, the calculations of the z-statistic and the p-value. A correctly-drawn graph helps.

The decision part must link the decision to the reason for that decision, citing the statistics and including the actual language of the problem. This means that you have to answer the question asked using the words provided in the prompt (the context). To make your answer *shine* include a well-worded statement that demonstrates to the reader that you really understand what the p-value means.

HW due Monday 2/8/2010 12.3, 12.4, 12.6, 12.13, and 12.14.

If you can't find time to do the homework, I will hold afterschool detentions to help you with the scheduling.

Standards IV A3, A5, and B3

Go to the AP Statistics documents page to download an example of bot a 2 proportion CI and a 2 proportion HT for the data collected in 6th period.

HW due 2/17/10: Problems 13.7-13.10 from the text.

Standard IV B1

Type I and Type II error, alpha, beta, and the power of the test.

Type I error: rejecting the null hypothesis when the null is actually true.
Type II error: failing to reject the null when it is false.

P(Type I error) = alpha. We have the privilege of selecting this value.
P(Type II error) = beta. We calculate this using the rejection region boundaries and the true distribution. This requires a new theoretical parameter.
Power = the probability that the test will be able to detect a difference between the hypothesized value and the new, theoretical value.

Power = 1 - beta

Beta = 1 - power

No formulas combine both alpha and beta.

Calculator method for computing beta: normalcdf(lower critical value, upper critical value, new theoretical mean or proportion, standard error).

For instance, the lower and upper boundaries of the "fail to reject" region if the hypothesized p is 40% and n = 200 are .3321 and .4679. What is the likelihood that we fail to reject when the true proportion is 48% (meaning that we can't distinguish between the 50% and 48%)?

std error = .035

normalcdf(.3321, .4679, .48, .035) = .3648. About 36.48% of samples drawn from the distribution with proportion = .48 will not make us reject the null hypothesis.

Error warning: If you get 95% when you make this type of calculation, you are probably using the original hypothesized parameter, and not the new theoretical one. Try again using the theoretical value.

And the really good news is that this method works for inferences for means and differences of means, too, so we don't have to learn another new procedure.

HW due 2/19/2010: Finish problem 13.30 a-d. This will take more than one page.

Answers for 13.30 should include the following elements.
A. Two treatment groups, random assignment(not random sample), first group of 1/2 people took only aspirin, the other both drugs.
B. Test statistic = 2.73. Complete answer requires all the rest of the HT work, including computation of combined p-hat.
C. (-.0232, .0197) with supporting work and interpretation on context.
D. Explanation of each type of error is required. II is more serious because of potential harm to patients.

HW due 2/22/10: Using part c of the 2009 AP exam question #5 as a guide, re-consider at least 5 of the inference problems we've already worked. "Based on your conclusion . . . which type of error, Type I or Type II, could have been made? What is one potential consequence of this error?" Write complete responses. Please pick problems that have each of the responses, reject and fail to reject, so you can get practice answering the problem both ways. Also, be ready to discuss the effects of Type I and Type II errors on HIV testing, pharmaceutical studies, and court cases.

The Central Limit Theorem and Sampling Distributions

2010-01-11T17:52:00.008-05:00

Standards: IIID 1, 2, and 3
Why is the Central Limit Theorem so important, even when the distribution of x is not Normal?

Key understandings:
When does the CLT "kick in?"
Why is the standard deviation of the sample averages smaller than the standard deviation of the population?
How do you apply the CLT to compute probabilities related to the sample average?
What does sample size have to do with your certainty about the distribution?
How do these methods apply to sample proportions?

Stay on top of the material by reading the chapter and taking notes on the key concepts. Work problems. We will test on Thursday, January 21st.

Here's a version of the sampling program that works!

-> means "store"
Lbl, For, IF, End, and Goto are all programming words, so they are found under the programming menu. Just hit PRGM for access to the menu.
DON'T TRY TO PROGRAM THE ITALICIZED EXPLANATIONS

:1 -> C *Initializes the value of C, the counter
:rand(150) -> L1 *Puts 150 uniform random numbers in L1
:Lbl 10 *Labels this line for use later
:0 -> B *Resets the value of B, the partial sum of the numbers, to be 0:rand(150) -> L2 *Puts 150 more random numbers in L2
:SortA(L2,L1) *Sorts your original numbers by the second column:For(J,1,10):L1(J)+B -> B: End *Takes the top 10 numbers, and adds them together
:(B/10) -> L3(C) *Calculates the average and puts it in the next row of L3:C + 1 -> C *Changes the counter to the next number
:If C < 101: Goto 10: *Starts over to select another sample of 10 from the population

Please answer these questions every time you work a problem that requires a calculation.

What is the population of interest?
What is the sample?
How can you justify using the Normal distribution?
Why are you allowed to use that simplified standard deviation?
Is your sample large enough?
Is your sample an insignificant part of the population?
Was your sample selected randomly?
Are the observations independent?

What is the distribution of the sample statistic?
Have you drawn the normal distribution graph and labeled it?
Have you shaded the appropriate part of the graph?
Have you checked your answer to see if it looks reasonable?

Work a bunch of problems from the text. Cici's 2-4 Sunday. Test Thursday.

Chapter 7 - Random Variables

2009-11-17T16:01:00.002-05:00

Compacting time! Look over the section summary for section 7.1. Unless you have questions, we will assume that you've already learned this part and we will move on.

Work problem 7.38 (and 7.37 for those who missed class today).

Make progress on your book. If you need a book approved, send me an email.

Chapter 6 - Probability

2009-11-04T15:40:00.003-05:00

Now probability wasn't so bad, was it?

Today we investigated multiple representations of categorical data: contingency tables, tree diagrams, and Venn diagrams. Each has its merits. All will provide the information you need to answer probability problems.

HW due Thursday, Nov 5: Take notes on section 5.2, pages 407-417 in the text. Work at least two problems from each of the problem sets in that section.

Standards: Section IIIA all
Concepts: Law of Large Numbers, multiplication rule, addition rule, sample space, continuous and discrete random variables, independence, expected value.

Essential questions: Why would we call the laws of probability laws? How can they be used? What does mathematical independence mean? How do we extract the important elements from a word problem so we can solve it?

Work problems from Chapter 6 in preparation for a test on Monday, November 16.

Chapter 5 - Producing data

2009-10-27T16:43:00.001-04:00

This unit covers survey design, observational studies, and experimental design. The standards involved are found under section II:
II. Sampling and Experimentation: Planning and conducting a study (10%–15%)
Data must be collected according to a well-developed plan if valid information on a conjecture is to be obtained. This plan includes clarifying the question and deciding upon a method of data collection and analysis.
A. Overview of methods of data collection
1. Census
2. Sample survey
3. Experiment
4. Observational study
B. Planning and conducting surveys
1. Characteristics of a well-designed and well-conducted survey
2. Populations, samples, and random selection
3. Sources of bias in sampling and surveys
4. Sampling methods, including simple random sampling, stratified random sampling, and cluster sampling
C. Planning and conducting experiments
1. Characteristics of a well-designed and well-conducted experiment
2. Treatments, control groups, experimental units, random assignments, and replication
3. Sources of bias and confounding, including placebo effect and blinding
4. Completely randomized design
5. Randomized block design, including matched pairs design
D. Generalizability of results and types of conclusions that can be drawn from observational studies, experiments, and surveys

We have looked at the mechanics used in selecting random samples using the table of random digits and simpler methods.

HW due Monday: Work as many problems from pages 371-373 as you need to be proficient with blocking and matched pairs design.

HW due Wednesday: an annotated vocabulary list from this chapter. Include explanations of why each term is good for design or a problem for design.

HW due Friday: Bring a printed copy of your electronic research proposal. We will be modifying it. You should go to the Cobb County School District website to see what the requirements are for research in our schools. Also go to the Institutional Review Board (IRB) site for the college you are most interested in and review their requirements. Be sure to answer all the questions these forms aske except for the statistical analysis questions.

Chapter 4 Non-linear relationships

2009-10-01T17:49:00.003-04:00

Standards: I D (exploring scatterplots, transformations to achieve linearity) and E (exploring categorical data, two-way tables, etc.).

Thursday 9/24 Today we revisited residuals and the LSRL. We looked at data that appeared at first to be linear, but upon inspection were clearly not linear. That's what residuals can do for you!

We also straightened our first data set. We took exponential data--ordered pairs of the form (x, ab^x)-- and transformed them into a straightened set. Once you have straightened data, you can use the LSRL function on the calculator. We found the LSRL, converted it to a curve using our knowledge of exponents and logs, and graphed the curve through our exponential data. Ooo. Ahhh.

Procedure: Enter x and y into L1 and L2
Look at the data. See that they are not straight, but exponential in shape.
Take the ln of the y values (put in L3).
Look at scatterplot of L1, L3. Straight? Then --> LSRL
Change y-hat to ln-y-hat because we used the ln y instead of y.
Solve for y.
Graph that new equation with the original L1, L2 data.
Be proud.

September 28th: We looked at several non-linear models and discovered what transformations would make the "right" side a linear function. Those realizations drive our decisions to take logs or square roots of the original variables.
HW: problems 4.11 and 4.12 from the text.

September 29th: We worked through parts of problem 4.12 and reviewed properties and purposes of logs. Do problems 4.15 and 4.16 for Wednesday.

October 1st: Worked with transformations more today. Finished up the analysis of the disappearing dice lab where we modeled exponential decay.
Took a quiz on residuals to give students an opportunity to recoup some points from the Ch 3 test. It worked for some. Why pass up an chance to improve your grade? A copy of one version of the quiz can be found on the Typepad blog. Scroll down to Documents for AP Statistics.

October 2nd: Quizzed again today on computing, graphing, and interepreting residuals. This concept is critical to continuing in Stat. Most students have now demonstrated mastery, but those of you who have not shown me that you can do it need to step up! HW due Monday: 4.26, 4.27. 4.28 from the text. Be prepared for the next quiz on finding residuals and transforming data.

October 5th and 6th: We've been spending a lot of time perfecting our understanding and skills regarding transformations, least squares regression, and interpreting residuals. We will have nearly daily quizzes to assess our progress. In addition we are looking at contingency tables (2-way tables). We computed joint, marginal, and conditional probability and took a quick look at the meaning of independence.
HW: Read section 4.2 and do problems 4.29 and 4.30. They are pretty cool.

Chapter 3 Exploring Linear Relationships

2009-09-11T16:40:00.003-04:00

No CiCi's on Sunday, September 20. Let's give the Accel Math II kids a chance.

The focus of this unit will be on standards I D 1-5 plus inference for regression (IV A 8 and IV B 7). These topics can be found in Chapters 3 and 15 of the text.

First up, we look at graphs of bivariate data. You must be able to graph (x,y) pairs on the Cartesian plane. We will be finding the Least Squares Regression Line (y-hat = a + bx) and interpreting multiple measures of fit. R is not the answer!!! You will be expected to calculate the LSRL using tables and formulas. There will be a couple of formulas that you should memorize, but now is a good time to get the green formula sheet out to see the formulas that you will be provided on all tests.

The most important formula so far is the formula for residuals: y - y-hat = the observed minus the expected [for each value of y].

Be prepared for a lab in class on Wednesday. Dress appropriately.

Friday 9/11/09: In our short period today we investigated the deviations and the residuals and just barely got to the formula that uses the squared deviations and the squared residuals:
First The deviations = the predicted differences + the residuals
Second 1 - (the sum of the squared residuals)/(the sum of the squared deviations) = r^2
Did you get that? R-squared equals the portion of the squared deviation that is not the squared error part. In other words, it is the part of the deviations that we could have predicted.
TO get these sums of the squared values we used the lists in the calculator like this:

L1: the x values
L2: the y values

Ran LinReg L1, L2, y1

L3: the predicted values of y FORUMULA= Y1(L1)
L4: the squared residuals FORMULA= (L2-L3)^2
L5: the squared deviations FORMULA= (L2-mean(L2))^2

Then use the LIST MATH 5.sum (L4) and (L5) to get the sums of the squared errors SSE and the sum of the squared deviations SST, respectively.

The R^2 formula is then 1 - (SSE/SST).

Take notes on all of Chapter 3 for Monday. Our test is Thursday.

The formulas we used in class on Monday were

b = r * Sy / Sx.

r = (sum of (Z of x * Z of y))/(n-1)

a = y-bar minus b * x-bar

HW is at least three and no more than 15 problems from the Chapter review for Chapter 3. We will cover the aspects of inference for linear regression on Tuesday.

s = the standard error about the line= an approximation of the average residual for that LSRL

SEb = the standard error of the slopes of the regression line. You would expect the slope to vary by about this much on average when you used different points from the population to come up with a LSRL.

beta = the slope of the real relationship between x and y, is approximated by b

alpha = the real y-intercept of the real relationship between x and y, approximated by a

Confidence interval for the slope : b +/- about 2 * SEb

T-statistic for testing Ho: beta = 0
b/SEb

STUDY for the test.

Test date 9/17/09.

Preview Chapter 4 for homework. See you on Monday.

Chapters 1 & 2

2009-08-28T16:19:00.003-04:00

We're tying up loose ends related to interpreting visual displays of data and describing or summarizing the distributions. Our test on topics of chapters 1 and 2 will be on Thursday, September 3.

Standards: IA1-4, IB 1-2

HW due Monday: Problems 2.10, 2.12, 2.13, and 2.14. Problem 14 requires you to use your TI-83 calculator.

See you at CiCi's on Sunday from 2-4. That's near the Super WalMart at Trickum and HWY 92.

Test on Chapters 1 & 2 is Thursday. Prepare by reviewing the Chapter summaries, using a study guide, taking the online quizzes at the textbook website.

Welcome to the new school year

2009-08-25T17:35:00.000-04:00

Welcome to AP Statistics.

For the first two weeks of school we will explore aspects of data collection, interpretation, and representation, topics identified by College Board in the Topics Outline for AP Statistics.

EQ: What challenges are there in collecting data? What options do we have for representing data? What changes would we make for different audiences?

8/10/09: Standards IIB 1 & 3
HW: replace the batteries in your TI-83, TI-84, or N-spire calculator, procure a bookcover to bring to class next week, and generate a lovely graph of the data you collected today.

8/11/09: See last night's homework PLUS create a graph that shows the cumulative frequency for the different responses, from most frequent to least frequent. The height of the first bar should be the frequency of the most common reponse, while the last (cumulative one) will be the size of the class.

8/12/09: Great job on the normal distribution today. Use it in good health in AP Psychology! If you haven't replaced your batteries, you still need to do that. If you haven't procured a book cover, then check into your many options. See you Thursday. Standards IC4, IIIC1

8/13/09: Today we got our textbooks and worked with the standard normal table. Do problems 2.23-2.28 from the text for homework. You will find these problems in Chapter 2 (of course!) numbered sequentially from the beginning of the chapter. There are examples and explanations in the pages leading up to these problems in the text, so use your resources wisely.

8/14/09: Does your brain still hurt? Today we explored probability density functions including the uniform, the Normal, and the triangular. When we looked at the triangular we reinforced our notions of percentiles and probability in the context of a continuous function. The standard introduced today was IB3. Your homework is problems 2.31-2.34 in the text. These are the 31st through the 34th exercises in Chapter 2.

I am not planning to start the CiCi's Sundays yet because we are still just surveying the concepts from the course. If you have questions, please see me before school on Monday. Have a super weekend. Go Lady Trojans Softball and Volleyball players! Good luck against Colquitt Co, football players! Marching band, march on with pride!

Please cover your textbook and CHANGE THOSE BATTERIES!

HW due Tuesday Aug 18th: Problems 1.7, 1.8, 1.9, and 1.11 from Chapter 1. The problems are numbered sequentially in the text, so start at the beginning and keep going until you find these problems. If you have difficulty, look at the examples on the pages prior to the problems.

HW due Wednesday Aug 19th: All the problems from the section 1.1 summary area (page 67-69) that you know how to do. I will assign additional work for each problem that you don't know how to do to get you caught up, so do your best.

HW due Thursday Aug 20th: Using the data you recorded in class today about work experience and gender, create and discuss the graph that best reflects the relationship between these two characteristics. Also, here's an interesting link about the ACT. Compare the performance of each subgroup to the national subgroup averages. Then compare the overall performance of Georgia students to the entire population of test-takers. Can you explain the difference?

HW due Monday, August 24th: Bring two different types of published graphs to class (cut out of the newspaper, magazine, or other publication or printed from the Internet PLUS enough information about the data collected that you can write up an interpretation as an expert.

HW due Tuesday, August 25th: Problems 1.39, 1.40, 1.42, and 1.43. Bring those graphs from the weekend, too!

HW due Wednesday: THe same as last night! Be sure to bring these items tomorrow. ALSO, cover your textbook.

NO CiCi's this week. Probably next!

Last few weeks

2009-05-04T18:36:00.004-04:00

QUESTIONS FOR JUNIORS AND SOPHS:
1st - 5
2nd - 2
6th - 6
7th - 5
Answer completely and correctly. Bring it with you to the exam along with a calculator and a pencil, but no notebooks or backpacks. We need to cut down on the end-of-year littering around the school neighborhood.

Senior take-home exam assignments: Bring a perfect and complete response to the assigned question on the day of your final along with your textbook and green t-table card. Typed is OK.
1st period seniors: Question 5
2nd period seniors: Question 4
6th and 7th period seniors: Question 1

The random assignment of questions for sophomores and juniors will take place on Tuesday.

DO NOT TALK ABOUT MULTIPLE CHOICE EXAM QUESTIONS. We can talk about the free response questions starting Friday. This means that you cannot write, text, or twitter about the questions, either.

Good luck on the test Tuesday! Please bring pencils, pens, calculator, tissues, and ID.
DO NOT BRING any books, notes, food, drinks, noisy things, cell phones, cameras, fireworks, explosives, plutonium-based products, live animals, fruits or vegetables, etc.

Plans for the rest of the year:
Recognizing that many of you will be taking other exams between now and the final, we will be working ASMA tests, reviewing adv algebra and trigonometry, developing rubrics for the questions you see on Tuesday, and preparing for the final.

Chapter 15 and other topics

2009-04-29T14:33:00.000-04:00

E-mail your edited chapter summary to jhl2881@students.kennesaw.edu as an attachment.

You will receive an invitation to join a private Wiki.

Do not put your name on the document, but DO put your name in the email title.

SENIORS IN 1st and 2nd PERIODS: Go to Room 808 for class on Wednesday.

Pre-testing processing of the exam forms for AP testing (otherwise known as the Bubbling-in sessions) have started. Go to the theater at 7:30 in the AM or 3:30 in the PM to fill in your forms in anticipation of the big day.

Next test: Cumulative multiple choice test on Thursday, April 16.
Test after that: TUESDAY, April 21. (Unless you sign up for Monday afternoon free response testing.)

We plan to test on Chapter 15 on Tuesday, April 14. Until then we will be doing problems from old exams that relate to linear regression and inferences about the slope (the problems you Hawaii folks picked up on Tuesday!!!). We covered this stuff first semester when we did regression.

Our tentative plans for tests for the rest of the semester. . .

4/14 Chapter 15
4/15 Sophomore testing in the morning
4/16 Multiple choice test #1
4/21 Multiple choice test #2
4/23 Multiple choice test #3
week of April 20-24 Free response test after school 3:45 - 5:15 one day
May 5 the big one!
Sometime later Final exam

HW due Thursday
Write up the responses to the three free response questions discussed in class in your spiral notebooks.

Notes for the 2008 problem, part c: Follow their directions. Average the two proportions. For the standard deviation you will have to find the square root of the variance of the average. Yiles! Sounds difficult.

Variance of the average = 1/4 variance of x + 1/4 variance of y.

Chapter 14 Chi-square procedures

2009-03-27T15:36:00.000-04:00

We have covered all the new material from this chapter. Your test will be on Tuesday, March 31.
HW for this weekend: AT LEAST three problems (not all odd) from the Chapter 14 chapter review. If you are struggling, do more problems. Bring all your CH 14 homework for credit on Tuesday.

Preview for after the test: We will be re-doing inference for regression (linear regression t-tests and confidence intervals) and power of the test.

Problems 11, 15, 16, and 24 are due Friday.

Problems 14.3, .4, .5, and .8 are due Wednesday.

HW due Tuesday: Using the data from your M&M bag, compare the sample distribution with the OLD theoretical distribution of BROWN 30%, RED 20%, YELLOW 20%, BLUE 10%, ORANGE 10%, and GREEN 10%. Perform a complete goodness of squares analysis of your data. If you misplaced your data, I guess you'll have to use a second bag of M&Ms. A complete analysis includes all the elements of SCAD.

Chapter 13 Two-sample tests

2009-03-18T19:00:00.000-04:00

We're starting with tests of differences between two proportions. Usually our hypothesis is that there is no difference. Of course, we could look for a difference! Testing on Thursday, 3/19.

Bring all your HW from this chapter to the test on Thursday.

Problems 46 and 48 are due Wednesday.

Problems 13.41-44 are due Tuesday. Also, review your rules for variances. The variance of the difference of two means is equal to the sum of the variances of the two means. What is the variance of x-bar minus 3*y-bar if x and y are independent? Know your stuff.

Problems 15-18 should be done for Monday. Also, complete the hypothesis test from class:
Ho: mu E - mu N = 0 vs Ha: mu E -mu N < mu =" 7.26" dev =" 6.94" n =" 100" mu =" 9.55" dev =" 5.88" n =" 100" color="#ff0000">Your test on this chapter is on Thursday of next week.

HW due Thursday: Problems from the book 13.2, 3, 5, and 27, PLUS complete SCAD write-ups of the hypothesis test and confidence intervals for the differences in proportions from the census data linked below. We are looking for the difference between the % of adult Americans who have graduated from high school for women and for men.

Interesting 2-proportion statistic: http://www.census.gov/Press-Release/www/releases/archives/education/000818.html

People in the Lassiter area are happier than most: http://www.ajc.com/health/content/health/stories/2009/03/11/states_of_happiness_georgia.html?cxntlid=homepage_tab_newstab

HW due Wednesday: write up a complete (SCAD) response to the activity in class today. PLUS, read and create an outline for the second section of chapter 13.

Chapter 12 Tests about means and proportions

2009-03-04T12:23:00.004-05:00

If you missed the test on Monday, please make it up Tuesday in room 214 at 7:20 AM. If you are involved in Model UN, please make it up in room 313 at 7:20 on Wednesday morning.

In this VERY SHORT chapter we practiced the methods that we use for simple tests in the real world.

Test is Monday!!! Please try out the online interactive software Crunch It that the publisher provides. Just search on Crunch It. It does a lot of the same things that Minitab does and in the same ways.

HW due Friday: Because the problems from last night were mostly odds, tonight's assignment involves evens. Do two even problems from the chapter review. Your test is Monday.

HW due Thursday: EITHER 12.9, 19, & 37 OR 12.3, 13, 23, 29, 31, & 34.
To get credit, all work must be shown. Copying out of the back of the book will not suffice.

HW due Wednesday: 12.15, .16, .18,. .24, & .26.
HW due Monday: Read through the first section, taking notes on the key differences between previous chapters and this one. Pay particular attention to the standard errors and the kind of test used as well as the changes to assumptions/conditions. Work problems 12.2, 12.4, 12.5, 12.6, and 12.12.

Chapter 11 Tests for significance

2009-02-23T15:58:00.014-05:00

SIGN UP FOR AP EXAMS!

Check THIS website for an article about some of our superstars of statistics.

The applet on this webpage should be enlightening regarding Type I and Type II error. Scroll down the linked page a bunch until you find the yellow box entitled Statistical Errors Applet. Read the information in the little yellow box and click on the link where it says Applet 1. Statistical Errors near the little graph to display the applet. Move the box in the scroll bar at the top to change the value of alpha (the probability of a Type I error). In real life you can change this value, but pick your alpha before you collect your data! Otherwise, you might be accused of manipulating the data and giving statisticiansa bad name.
Now, move the middle of a NEW and IMPROVED distribution by sliding the box in the scroll bar at the bottom of the window. See what the yellow region looks like when you overlap the distributions. The yellow area represents the probability of a Type II error.
So, what effect does changing alpha have on the probabilty of a Type II error? When is beta maximized? When is it minimized?

TYPE I and TYPE II errors
PLEASE read the section in the book regarding these topics.

You will only be able to calculate a POWER or a BETA (the probability of a Type II error) when some NEW mean is introduced. The power of the test is the probability that the test will be able to distinguish between your original hypothesized mu and the newly proposed mu. The probability of an error is BETA.

To calculate BETA:
Find the boundaries of the FTR region for your original hypothesis. Find the probability that x-bar would fall between that lower bound and upper bound GIVEN the NEW mu and standard error of the mean. In calculator language [that you would NEVER write on a test] it would be normcdf(LB, UB, NEWmu, sigma of x-bar).

HW due Wednesday: 11.36-11.40. Your test is Thursday.

You should have worked problems 11.5, .6, .27, .28, .29, .30, .49, .50, & .51 by Tuesday. Your test is Thursday.

The excerpt in class today was from The Lady Tasting Tea by David Salsburg.

HW due Thursday: 11.3, 11.4, 11.6

Please note that (1) null hypotheses ALWAYS have an "equals" concept
(2) null and alternative hypotheses do NOT include statistics.

In inference testing, the results of our sample may make us reject the null hypothesis if they are so unlikely that they would be unbelievably unlikely due to randomness.

Please read through the top of page 693 AND register for the AP exam.

Chapter 10 Confidence Intervals

2009-02-04T13:10:00.015-05:00

Your test on Chapter 10 will be Tuesday, February 17.
HW due Friday: Either problem 10.53 worked out in detail showing all work or problems 10.54 and 10.58 worked cout completely.
Have you hugged your study guide lately????
HW due Thursday: Problems 10.38 and .44. 1st and 2nd periods, please bring all HW from this week on Thursday.

Today we computed paired t confidence intervals for the difference in grip strength between right and left hands.

You can find the t* value for any number of df by using the calculator;
STAT TESTS T-INT Stats x-bar = 0, sx = sqrt df+1, n = df+1, conf level = whatever you need, like .95.

ALL students should have finished 10.28, 10.30, 10.31, and 10.31 PLUS the summary of the cautions. Have these with you on Wednesday.

HW for 1st and 2nd periods: Summarize the cautions of section 10.1 (pages 635-637) in your own words and work problems 10.28 and 10.30.
Periods 6 & 7: Work problems 10.7-10.10 PLUS summarize the cautions above.

The question was raised: Why do we use 2 sometimes and 1.96 other times for Z*? As you probably recall, approximately 95% of the data in a Normal distribution will fall within about 2 standard deviations of the mean, but that was just an estimate. The more precise number of standard deviations that form the 95% boundaries is 1.96. Use that whenever we are using Z procedures UNLESS we are just looking for a quick and dirty estimate. but NOT when we are constructing confidence intervals.

When do we use sx and when do we use sigmax? Sigma represents the population standard deviation, a number we rarely know. On the other hand, sx represents our sample standard deviation. When we do not know the population standard deviation we will use t procedures instead of z procedures.

And, of course, we divide by sqrt of n to convert these standard deviations into standard errors of x-bar.

Some web-based applets for Confidence Intervals: Rice Univ Freeman

HW due Friday--
1st and 2nd per: Problems 10.7-10.10 from the text. You should have already worked the problems from the REVIEW III on pages 610 and 611.
6th and 7th per: "Review III" questions following Chapter 9 on pages 610-611 in the text AND print one page from a confidence interval applet from the web and be able to explain it.

Key concepts from today: Approximately 95% of sample averages will fall within about 2 std dev/Sqrt(n) of the population mean. If we don't know what the population mean is, we might reason that our point estimate (x-bar) is a pretty good guess, and that 95% of the time, our sample averages will fall within 2 std dev/sqrt(n) of the true mean. Then the interval (x-bar minus 2*std dev/sqrt(n), x-bar plus 2*std dev/sqrt(n)) is our confidence interval or reasonable guess at the value of the population mean. About 95% of these intervals will capture the true mean. The distance from the mean to the upper bound (or event the lower bound) is the margin of error.

These are NOT true: 95% of the time this interval contains the mean. 95% of population means fall inside this interval. 95% of the time the mean falls between lower bound and upper bound. NONE of these are true, so DO NOT write these as interpretations of the confidence intervals.

Instead, we are 95% confident that the mean falls between the lower bound and the upper bound.

If you did not work problems from the review following Chapter 9, now is the time!!!!

Chapter 9 Sampling Distributions

2009-01-30T14:57:00.002-05:00

If you missed the test on Tuesday, take it Tuesday PM at 3:30 in Mrs. Prestwood's classroom. HW for tonight: Do at least 5 of the 10 problems in REVIEW 3 which follows Chapter 9. Get ready for confidence intervals!!!!

http://www.stat.sc.edu/~west/javahtml/ConfidenceInterval.html

Prepare for the test by working problems from the text and by using a study guide (if you have one) to practice with multiple choice problems. You are welcome to come by in the morning to use a study guide in the classroom.

Your test on Chapter 9 is Tuesday, snow or no snow. Work lots of problems from the chapter. Be sure that you know how to check assumptions or conditions. Do you know when you are calculating probabilities for means and when you are calculating probabilities for proportions? You have to use the right conditions and formulas or you won't be answering the question.

Oh yeah, CiCi's Sunday. Super Bowl Sunday.

For Thursday, 1st and 2nd periods: Complete the questions from the AP exams that we looked at in class. The first one asked for (1) the probability that a measurement of a depth of 2 was negative when the error of the measurement was Normally distributed with mean 0 and std dev 1.5.
(2) What is the probability that at least one of three independent measurements from this distribution was negative?
(3) What is the probability that the average of three independent measurements from this distribution was negative?

Everybody needs to work problem 3 from the 2007 exam.

HW Problems 9.20, .21, 25., .26, &.29 due Wednesday.

HW Problems 9.19, 9.27, 9.30 due Tuesday.
Summary of the three sections of the chapter:
A sampling distribution is the distribution of the sample means of all possible samples of size n. As n (the sample size) increases, the variability of the sample means decreases.
When the underlying (original) distribution is Normally distributed, the sampling distribution for samples of any size n will be Normally distributed.
When the underlying (original) distribution is NOT Normally distributed, the sampling distribution for large sample sizes will be approximately Normally distributed. The closer the original distribution was to Normal, the smaller the sample size required to make the sampling distribution approximately Normally distributed.
These concepts can be applied easily to two cases: measures of x and sample proportions.
For measures of x: The mean of the sampling distribution of x bar is the mean of the underlying distribution of x. The standard deviation of the sampling distribution of x bar is the standard deviation of the underlying distribution /the square root of n.

For sample proportions: When np and nq are both > 10 and n is less than 1/10 of the population, the mean of the p hats is p and the standard deviation of the p hats is the square root of p*q/n AND the distribution of p hats is approximately normal.

< Link to a history of the penny.
Link to a more official history of the penny.

HW Problems 9.32 and 9.34.

I will NOT be available at Open House Thursday night. Please email me with any concerns or join us at CiCi's on Sunday.

HW due Friday, 1/23: 9.10, 9.12, 9.14. Students from per 1 and 2, email your averages to Mrs. L if you did not load them in class. Periods 6 and 7, look up the phrase "planned obsolescence."

This chapter requires you to recall some vocabulary from previous chapters.
HW due Thursday, 1/22: Problems 9.1, .7a-e, .9, .11, .13. You should read through the sections in order to understand the questions.
Also, periods 1 and 2, bring five results from mean(randBin(100,.5,100)).

Measures of central tendency
Median
Mean
Mode

Measures of dispersion (spread)
Range
Standard deviation
Variance
Interquartile range
Absolute deviation

Graphical displays
histogram
line graph
stem and leaf
box and whisker graph
probability density function
scatterplot
cumulative density function
dot plot
pie graph
bar graphs

Pictures speak louder than words

μ = population mean x bar = sample mean (unbiased estimator of mean)

If a sample is drawn at random from a population, the mean of the sample is an excellent estimator of the mean of the population.

σ = population standard deviation s = sample standard deviation

Recall that the calculation of s requires division by n-1 for some complicated reasons.

sx is the standard deviation of the distribution of x
is the standard deviation of the means of the samples of x

E[x] = E[ x bar]

Chapter 8 and the new semester

2009-01-14T16:00:00.000-05:00

HW due Thursday: Problems 8.41, 8.43, and 8.44 from the text.

HW due Wednesday: Problems 8.45-8.50. Your test is Tuesday of next week--the day after the Dr. King holiday.

How are these questions similar? How are they different? What strategies would you use to answer each?

1. Of the 20 cell phones in a classroom, 30% do not accept text messaging. What is the probability that 3 out of a sample of 7 drawn from the 20 with replacement will not accept text messaging?

2. Of the 20 cell phones in a classroom, 30% do not accept text messaging. What is the probability that 3 out of a sample of 7 drawn from the 20 WITHOUT replacement will not accept text messaging?

3. Of the 200,000 cell phones in a metropolitan community, 30% do not accept text messaging. What is the probability that 3 out of a sample of 7 WITHOUT replacement will not accept text messaging?

HW due Tuesday, January 13: Probloems 8.19-8.24.

HW due Friday: 8.7, 8.10, 8.11, 8.13, & 8.16.
By Monday, make sure that all of the assigned homework has been done correctly and completely.

HW due Thursday: Problems 1-6 of Chapter 8. Each question requires that you explain how the 4 characteristics is satisfied. Also, define x in each setting. KEY: If your are not counting the number of successes (x) in n trials, it can't possibly be a binomial. If is IS, then check the rest of the conditions.

Yippee! We made to the home stretch.

We will begin Chapter 8 on Wednesday. Tuesday's HW is to complete as much of the crossword puzzle as possible. Some of the answers will become clearer as we progress through binomial and geometric distributions. The plan is to test on Chapter 8 next week and have a chapter test every 2-3 weeks thereafter. This way, we'll be able to dedicate the time after the break to review for the exam.

Let's see. Remove Mirage. Replace batteries. Ask parents to read and sign the syllabus. Bring paper, pencil, and calculator on Wednesday. Be safe.

Did I forget anything? :)

Chapters 6 and 7

2008-12-09T16:44:00.000-05:00

Work at least 6 challenging, interesting problems from Chapter 6. If you don't choose challenging problems, then you're not going to be prepared for Thursday's test!

*** The Gator cheerleaders came out to practice with the band tonight and, when they were done, went into the gym and cheered for the freshman basketball team. Go Gators***

And they just mentioned both Venn diagrams and Simpson's Paradox on Numb3rs!

HW due Monday, December 8: Problems 7.55 and 7.58. The test will be on Thursday.

HW due Wednesday, December 3 - 7.32, .37, .38, and .39

HW for Monday- Problems 7.1-7.3 from the text CHAPTER SEVEN!

Please be safe. I will not be at CiCi's on Sunday, November 23. Also, please note that the email system is going to be down until we get back to school in December.

Have a happy Thanksgiving.

Chapter 5 - Generating and collecting data

2008-11-17T17:45:00.001-05:00

Here's a website with some summary information about experimental design.

HW due Wednesday, 11/19/08: Problems 2006-5 and 2004-2 from the AP exams. Do the work on a separate piece of paper. No credit will be given for work done on the handout. If you missed class on Tuesday, swing by before school to get a copy or go to the AP website to retrieve a copy, but be prepared for class! Your test is on Thursday.

HW due Tuesday, 11/18/08: Problems 5.46-5.49. Your test on Chapter 5 is Thursday.
HW due Monday, 11/17/08: Problem 5, part c from the 2005 exam. The original question is found at www.collegeboard.com/apstudents.
A survey will be conducted to examine the educational level of adult heads of households in the United States. Each respondent in the survey will be placed into one of the following two categories:

Does not have a high school diploma
Has a high school diploma

The survey will be conducted using a telephone interview.
(c) Since education is largely the responsibility of each state, the agency wants to be sure that estimates are available for each state as well as for the nation. Identify a sampling method that will achieve this additional goal and briefly describe a way to select the survey sample using this method.

Your answer must be written and complete and reflect your best reasoning.

Your test on this chapter is Thursday.

HW due Friday: 5.9, 5.10, 5.13, 5.14. Know how each of these types of bias occurs and what its implications are: response, non-response, voluntary response, wording, undercoverage. Also, how does convenience sampling lead to bias?

Know also how to conduct surveys using each of these methods:
Stratified random sample: Create layers of homogeneous groups and randomly sample from each.
Systematic random sample: Select every nth person for the survey.
Cluster sampling: Survey a smaller heterogeneous group that has characteristics similar to the entire population.
Simple random sampling of size n: (the ideal) use random methods that ensure that every person has the same likelihood of selection AND that every possible group of n people has the same likelihood of selection.

Your HW due Wednesday 11/12 is 5.1-5.5 (after reading the first four pages of the chapter).

Here are the questions that will be on the long form and the short form of the census in 2010. The actual date of the census will be April 1, 2010.

Chapter 4 Examining Nonlinear Relationships

2008-11-06T16:30:00.001-05:00

Due Friday 11/7/08: Problems 4.41-48 in the book. YOu'll need to read the section that precedes these questions. The test is on Monday. Don't wait until the last minute to work problems. [Check out the Barron's sudy guide, sections 4 and 5 for more problems.]

UPDATE: The Chapter 4 test has been postponed to Monday, November 10, but you still need to work problems!

HW due Wed 11/5/08: You should have worked at least 5 problems from the first section of the chapter plus problems 4.23 and 4.26. It would be helpful for you to work more problems, since your test is Thursday.

HW due Friday: Bongiorno! Problem 4.12, the pizza problem. Sorry for the delay. The CCSD filters wouldn't let me post from school. si I had to wait until I got home.

HW due Thursday: problem 4.5 plus at least one of 4.6, 4.8, and 4.10. Finish your worksheet from today.

HW due Wednesday: Write out comprehensive notes for Chapter 4.

The test will be the Thursday after the election day.

Chapter 3 Examining Relationships

2008-10-21T15:35:00.004-04:00

This chapter introduces methods of describing, evaluating, and predicting a relationship between two variables. We are extending our look into Chapter 15 as well.

The test is Monday over Chapter 3 and Chapter 15. Most of the emphasis is on Chapter 3. How could you prove that you know the material?

Due FRIDAY, 10/24: Create detailed written instructions for finding r, the LSRL, residuals, s, and SEof b, performing the hypothesis test for a slope, and for finding the confidence interval for a slope.

15.6 and 15.8 from the text.

Due Wednesday, 10/22: Find 4 examples of regression output using a computer program in the text. For each one, construct null and alternate hypotheses for inferences about the slope of the LSRL, perform the calculations related to your hypotheses, and write out the conclusion in the context of the problem. We are ignoring for now the assumptions/conditions portion of the required steps in inference. All of this must be written on paper-not on your calculator or stored in your heads.

Due Tuesday, 10/14: Translate the definitions from Chapter 15 into language you understand more easily.

Due Monday 10/13: Problems .33, .34, .38, .39, .43, .44, .45. Problems 3.29-3.32 were due Thursday, 10/9.
Due 10/6: Become experts. It's up to you, now. Become able to calculate r, b, a, and the residuals in tables quickly and efficiently. Y'all need to practice so you are fluent!
It boils down to becoming experts in the same thing we've done on the calculator, Fathom, and Excel for 3 days. You could ue a data set from an example in the book and rework it until it is second nature.

Due 10/3: Work 2 challenging problems from the first section that you haven't already worked to PROVE that you are an expert on correlation.

10/1: Looked at data and linear relationships using Fathom. Become an expert on the first section of chapter 3 (3.1).

HW due Wednesday, 10/1: 3.13, .17, .19, .20, .23, .24, .28

Homework for 9/29: familiarize yourself with the correlation and regression applet on the textbook website.
http://bcs.whfreeman.com/tps3e/

HW due 9/26: Problems 3.5 and 3.6 from the text. Problem 3.5 involves one of my favorite relationships: manatee deaths cause powerboat registrations.

HW due 9/25: Be prepared to generate a linear model to solve the copier repair problem and defend your choice of variables.

HW due 9/24: Create an annotated outline of the chapter (include vocabulary and formulas and where you found them). This should take about 2 pages.

Chapter 2 - Describing Location

2008-09-18T15:00:00.001-04:00

The Chapter 1 tests have been graded and returned to students.
The test on Chapter 2 is Monday. Bring your calculator. C U at CiCi's if you're up for it! (Behind the Starbuck's on Hwy 92) Have you tried the online quiz?

HW due Friday, Sept 19 (International Talk Like a Pirate Day!!) Work at least seven challenging problems from Chapter 2 that we haven't assigned yet. The more you challenge yourself, the better prepared you'll be for the test on Monday.

HW due Wednesday: Problems 2.51 through 2.54. Chapter 2 test will be on Monday.

HW due Tuesday, Sept 16th: 2.46, 2.47, & 2.50.

HW due Monday, Sept. 15th: 2.43 and 2.44.

Come by before school to put CtlgHelp on TI-83 plus and TI-84 calculators.

HW due Sept 12th: problems 2.37, 2.38 AFTER you've read pages 148-154

HW due Sept 11th: problems 2.29, .30, .32, and .34.

HW due Sept 10th: problems 2.23-2.26 from the text
Health alert: http://www.reuters.com/article/newsOne/idUSSYD5846120080815

HW due Sept 9th: Problems 2.5-2.8 from the text.

HW due Sept 8th: Problems 2.10 and 2.13 from the text PLUS find the median, Q1, and Q3 of the triangular distribution defined by the segment connecting (0, 1) to (2, 0) or whatever triangle distribution (with the fat side on the left!!!) we assigned in your class.

Be safe.

Chapter 1 - Exploratory data analysis

2008-09-02T16:24:00.000-04:00

We are learning how to create graphical representations of data and to interpret different graphs.

MESSAGE TO AP STUDENTS: It is impossible for you to make up the discussions and interactions that you miss when you are absent from class. Therefore, it is your responsibility to keep up with the homework so you are not behind when you return to class. This means that you need to get the assignments from the blog or another student. Email me if you need more information--but don't show up empty-handed after an absence! You'll only get further behind.

HW for 9/2: Take notes on timeplots and ogives from the text. You will be expected to construct these graphs tomorrow in class. You would benefit from looking through the chapter summary to see if there is anything you don't understand/can't do. The test is THURSDAY!

HW for 8/29: Using the data collected in class, recreate the box and whisker plot, the stemplot, and the histogram representing the data. Look in your book for directions for the following: Back to back stemplots, back to back histograms, and parallel boxplots. Separate your data into boy data and girl data and create each of those graphs to compare the boy data to the girl data.

ALSO, consider the effect of outliers on the mean, median, standard deviation, and interquartile range of the data. Which of these measurements will be affected tremendously by outliers?

Be safe.

HW for 8/27: Bring in a graph that excites you. It must be a histogram, boxplot, stemplot, or other graph that you can write about BUT NOT A BARGRAPH. The graph can be from the newspaper, a magazine (with permission from the owner to cut it out!) or printed from the Internet.

HW for 8/26: Using the data from problem 1.11 in the text, create a split stem-and-leaf graph of the ages of the Presidents at inauguration. Write a few sentences describing what you have created and compare/contrast the results with the histogram that you created in class to represent the same data.

New Year. New Faces.

2008-08-21T16:00:00.000-04:00

Look for your homework below. Accel Math is in red and AP Stat is in blue.
Here's a treat for those who made it this far. Watch for the back flip and the dismount.

Calculator news! Check out these deals:
http://forums.slickdeals.net/showthread.php?t=874733

Welcome to the first posting of the new school year. Until the school website is updated with the new course, both AP Statistics and Accelerated Math I will share this blog. Both groups should change the batteries in their calculators.

Accelerated Math I- No HW for Monday 8/11. We generated some data today and began to analyze the class results. We'll continue to analyze the results tomorrow and look at data that are not discrete. Our primary topics for these few days center on functions and independent and dependent variables.
The Accelerated Math assignment for Tuesday night 8/12 is to complete the worksheet handed out in class (Fiona #1). For Thursday 8/14 night, students should complete problems 2 & 3 from Fiona worksheet #2.

Over the weekend, (8/15) make sure that you have finished problem #3 from the Fiona worksheet (using the results from problem 1!!!). Be careful. There are multiple parts to each section. Describe your method and the meaning.

HW for 8/18: Complete the yellow sheet: Problem #5. Be sure to answer all the questions asked.

HW for 8/19: Do as many of these as you need to become proficient--
Make up two ordered pairs of numbers. Calculate the slope of the line through those points. Calculate the y-intercept of the same line. Write the equation of the line through those points.

Hints: You can use the slope formula m = (y2 - y1)/(x2 - x1) to calculate the slope and the slope-intercept or point-slope formula to determine the y-intercept.

For 8/20:Finish the salmon (orange) worksheet you picked up in class. Answers should be on your own paper. The quiz results were mostly good. Most common error: graphing a line instead of dots when the domain only contains discrete points. We have to pay attention to these details.

For 8/21: Graph y = x-squared, y = 2 x-squared, and y = x-squared + 2 on the same graph we started in class. Use colored pencil if you can to differentiate among the graphs.

Calculator recommendations: The ideal calculators for Accel. Math I, Accel Math 2, and all higher math courses (including AP STAT) are the TI-83 and TI-84 families of calculators. The silver editions are preferrred, because they have more memory and more spaces for useful programmed applications. The TI-nspire calculator also has these capabilities and more. For an extra $30 or so you get something akin to a hand-held computer.

The TI-86 and TI-89 do not have the same sets of functions that the 83-84-nspire calculators do, and are not as useful in these courses.

AP Stat- We began to interpret an interesting graph. We'll continue to analyze the given graph and to generate some additional data tomorrow. HW for Monday 8/11: Describe in detail how to tie a shoelace. Tues 8/12: write up the directions for entering data (ordered pairs) into your calculator, graphing a scatterplot, and generating the best fit line. You will need to know how to perform these actions for the labs on Wednesday. For Thursday 8/14: Compute the number of days until the beginning of the next millenium showing all work. Explain your calculations and assumptions as you go.

AP Stat students - HW due Monday Aug. 18: Select 4 negative and 4 positive numbers with decimal parts between -6 and 6. (For example, -1.96 and 1.645). We will call those z-scores. Knowing that the mean (mu) of IQ scores according to a particular test is 100 and the standard deviation (sigma) is 15, compute the IQ scores that correspond to the eight z-values you chose. Check your work using the formula z = (x - mu)/sigma. . . also known as difference on the top and the error on the bottom.

Calculator recommendations are embedded in the Accel Math I information above.
Be safe.

HW for 8/18: Find the estimated IQ scores of 8 celebrities. Convert the IQ scores to z-scores. Of course, show the celebrity's name, his/her IQ score, and the z-score for that measurement.

HW for 8/19: (1) REWRITE the example that we used today in class, cleaning it up and adding explanations to the example. You should have both the old version and the new version in your notes. (2) Select a set of numerical data with at least 10 values. Using the notes as a guide, calculate the mean and sample standard deviation of your set of data by creating columns as shown. Create a histogram of your data.

HW for 8/20: find all the combinations of five integers [0,5], with minimum average, with maximum average, with minimum standard deviation, and with maximum standard deviation. First assume that numbers can be repeated, then do the problem again with no repeated numbers.

HW for 8/21: Find a site online with the rules for logarithms and re-learn them to be prepared for the mental exercises of Friday.

Click here for a cool site for histograms.

Click here for a decent site that shows how to calculate standard deviations.

Back from the AP reading

2008-06-20T10:42:00.002-04:00

The grading of the exams went smoothly, with the entire set of 110,000 exams graded by the close of business on Wednesday night. Although students from other schools seemed to think that the problems were hard, the teachers and professors who graded the tests were not complaining about a lot of low scores. I guess that's a good sign!

The chief reader (Christine Franklin, from UGA) evaluates the distribution and sets the cut points for the different final scores based on how students performed on the repeated questions from previous years. That done, the scores have to be approved by the AP psychometric gurus and processed for release to students. In other words, your scores should be available on time.

My role at the reading was a little different this year. In past years I've been assigned to a grading group of 12 people headed by two table leaders, then trained to grade two problems from the regular exam according to the rubric. For seven full days the people in each group grade those two problems. For instance, a reader may grade question #5 (the moose problem) and the table leader checks the scoring. Since there are six problems on the test and this year over 500 readers, there are a lot of people grading the same problem. Once a problem has been graded, that test booklet is bundled up and passed to a group that grades another question, and the process continues until all of the problems have been graded. If a reader has a question, she asks her grading partner, then her table leader. If the problem is not yet resolved, the question goes to the question leaders.

This time Ms. Franklin put me in a grading group with a bunch of REALLY experienced readers (famous and incredibly important people in statistics, like one of the authors of a popular textbook) who graded the make-up exam. We also graded two questions each (plus one from the regular exam), but we were the only ones to grade those problems. When we had questions, the people who wrote the rubric were right at our tables to answer and help us resolve any issues. Since none of you took the make-up exam this year, I never had to worry about accidentally grading one of my own students' tests, but every time readers commented on how great an answer was I told them that it must have been one of MY students. :) The trick with grading the make-up exam is that we can never divulge the contents of the exam to anyone, but that shouldn't be a problem because I have already forgotten it all!

You may be wondering (as I did) how I qualified for that special grading group. I think that they chose the very best former table leaders and other important people they could find, then filled in the rest with some experienced people from Georgia. Clarification: some of the important people were also from Georgia, so the line between the superstars and the poser (me) was blurred. Oh yeah, and they said that we we were selected because we were fast graders. I thought you'd enjoy that.

A few students, usually those who are stumped by a problem, write notes to the graders asking for mercy or generosity in grading. I'm sure they're joking, but they might be surprised to know how wonderful, helpful, and patient the graders really are. About half of the graders are college instructors who give up a week of their summers or coveted summer courses to validate this process for our students. The high school teachers are the ones who have attended or led workshops and conferences in statistics. Some of them haven't even ended their school years yet and have to pay their school districts back for the substitutes they needed. These people write the creative lessons published in books and shared online. They write the the textbooks and the study guides. They do it for students. They spend all day grading exams, then come back after dinner to hear a guest speaker talk about mathematical modeling or to share best practices. They are on constant lookout for ways to help their students understand statistics better.

This goes for the English Lit, French, and APUSH graders who were there at the same time, too.

I am honored and humbled to be invited to participate in this process with these amazing people.

Video time: http://www.youtube.com/watch?v=zTPhZcxnBSE&feature=related

http://www.youtube.com/watch?v=Ac3CtHYo5rY&feature=related

http://www.youtube.com/watch?v=IKkBBVxI5qU&feature=related

Final exam preparation

2008-05-19T17:22:00.002-04:00

(1) All classes have received their take-home portion of the final exam. The problem must be worked out completely, correctly answering the questions posed , on a separate sheet of paper. For underclassmen, the graph may be drawn on the question sheet itself. This is an individual assignment. You are not to collaborate on this problem. The problem is due on the day of your final.

(2) The in-class part of your final will consist of 21 problems from the practice multiple choice tests. The content relates to second semester topics.

Show me your best work!

The practice tests

2008-05-04T08:00:00.003-04:00

For Tuesday: Dress in layers because the gym is CCOOOOOOOOOOOOLLLLLLDDDD!

Replace your batteries if it has been a while since they were changed. Remember to run CtlgHelp (an APP) and Diagnostics ON (use catalog to find this).

Bring your student ID, pencils, and a pen.

Come to the outside classroom a little BEFORE THE END OF third period. You need to eat before you test. I'm bringing food.

Estudiantes del AP Espanol: We will save food for you. Come out to the outdoor classroom after the other exam.

DO NOT BRING study guides, textbooks, cell phones, i-pods, etc.

Post any questions to me by 9:00 for best shot at getting an answer.

GO TROJANS.

It's about 5:40 PM and I'm finally home. Please call if you have an urgent need to meet at CiCi's. 7/ 354 . 1791 (number is broken into pieces to discourage phishers).

I WILL NOT be at CiCi's at 2:00. You folks can still go and be an amazing, productive group, though. Be smart.

I will touch base again when I re-enter the cyper-world.

There will be ONE MORE multiple choice test on Thursday. The format is the same as the previous tests. This grade will replace the lowest of the multiple choice test grades. Be prepared.

The practice tests have been graded, the grades posted, and the adjustments are complete. There are three grades representing the original test score or a measure of improvement as follows:

If the original score on test 2 > test 1, then the original test 2 score is inserted as test 1's score as well.
If the score on test 3> test 2, then test 3's score is inserted as test 2's score as well.
No, if test 3 > test 2> test 1 you DON'T get test 3 posted three times.

Congratulations to those who scored over 100% on test 3. The highest score was 33 correct with 2 wrong for a raw score of 32.5 and a posted score of 162.5. That score also replaced the student's lower test #2 score.

OK. Have you ever had something in your eye and you just couldn't bring yourself to cry to wash it out? This would do the trick:
http://youtube.com/watch?v=WIxtxwiqK-s

I warned you.

Find some time this weekend to do the practice exam in the packet handed out on Thursday, but your next assessment is MULTIPLE CHOICE. Practice with a study guide. Find the topics that you don't remember. Brush up. Take a deep breath. See you Sunday for pineapple pizza?
- - - - -
Today's multiple choice test scores are posted to I-Parent. The next test will be Thursday.

The third opportunity for the practice free response test is Wednesday afternoon.

Video alert: The movie Wargames (released 25 years ago!!) is on amc right now. The best part is toward the end of the movie, when the computer gets smart.

Chapter 14 - The final frontier

2008-04-15T17:02:00.002-04:00

Inferences for regression

HW for Monday: Entire free response test from 2004. Take 90 minutes under test conditions.

HW for Friday: Problem 6 from 2007 AP exam.

HW due Thursday: Problems 14.11 and 14.18 PLUS the chapter summary.

HW due Wednesday: Problem 2 from the 2006 exam. Language is critical. If you do not know what the question asks, look at Example 14.5 in the text. Respond IN CONTEXT. There will be no partial credit for this problem.

DO NOT FORGET that your one-page summary of the selected chapter is due Thursday.

This chapter explains how to use the slope of a LSRL calculated from a sample and the std dev of the slopes

to create a confidence interval for capturing the slope of the true equation through ALL the data and
to make decisions about a hypothesized linear relationship.

Your calculator will do most of the number crunching. Investigate the LinRegT-Test function in the STAT TESTS area.

Why are there only n-2 degrees of freedom?
Why do we make such a big deal about s sub-b1 being the std dev of the slopes?

HW for Tuesday: problems 14.1 and 14.2 from the text.

Chapter summary due Thursday.

Chapter 13 Inferences using Chi-square

2008-04-02T14:15:00.000-04:00

Your Chapter 12 tests are available Thursday morning for pick up in room 313. I have posted an answer key outside room 313 for comparison.

What can you do to prepare for Thursday's test? Have you taken the online quiz? Select the link to the right>>>>>>

Have you used a study guide? You have now learned almost everything from the syllabus for the exam, so get to work!

Have you read the chapter in the text? Y, M, and M explained these concepts differently. If you didn't "get it" the way I explained it, you might like their version.

Did you google lesson chi-square test or something like it? Sometimes you can find nifty applets that make the concepts clear.

See you on Thursday.

HW due Wednesday, April 2: Complete 5 problems from the chapter review (NOT 28 or 29) in preparation for Thursday's test.

HW due Tuesday, April 1: Read pages 717 through 724. Do problems 13.14 and 13.16.

HW due Monday, March 31: Problems 28 and 29 (which requires simulation). Leave yourself enough time to do problem 29 or go to CiCi's.

HW due Friday: Problems 13.9, 13.10, 13.12.

HW due Thursday: Write up your answer to the free response problem handed out in class AND work problems 13.1 and 13.3.

We will study three different types of test that use chi-square methods in this chapter:

goodness of fit: compares one sample distribution to an expected distribution to see if it matches the expected counts closely or not (for example, comparing a sample of M&Ms by color to the % the company says it produces).

test of independence: determines whether there is evidence that a relationship exists between at least two categorical characteristics within a population based on the counts of observations in one sample (for example, analyzing applications to law school to see if undergraduate major and admission rate have a relationship).

test of homogeneity: compares counts within the distributions of two or more comparable samples to see if there is a significant difference inthe mixes (for example, comparing color distribution of a sample of milk chocolate M&Ms to the distribution of a sample of peanut M&Ms to see if there is a difference).

Each of the methods uses the formula sum [(O-E)^2/E]. This is the formula that Dr. Chuck Tate had written on his "grill" in the Statz 4 Life video.

Our plan is to do lots of labs and practice writing free response answers for the next few days. Our test is Thursday of next week.

For more information, scroll down to the Chapter 13 information from 2006 and 2007 on this blog.

Our probability lab is a week from Friday. We need parent volunteers to help with the activities. Please ask your parents to email me if they want to be part of this fun experience. They'll have a blast.

Chapter 12 Inference for proportions (Yeah!)

2008-03-22T08:30:00.000-04:00

HW for Monday: In preparation for Tuesday's test, the MINIMUM assignment is to work four problems (your choice) from the Chapter Review. This means that at least four problems have to be completed correctly. For instance, if problem 28 had been in the review section, your answer would include a complete hypothesis test or confidence interval--not just your opinion.

HW for Friday: 12.25, .26, and .28.

March Madness: post your NCAA men's basketball brackets at CBSSportsline with the Linnerstats bracket. The password for the bracket is just what you'd expect, gogators. If you are a bracket rookie, this means that you predict the winners of the 63 games in the NCAA basketball championship tournament. The software at CBSSportsline makes it really easy. You have to work fast. . . the brackets close soon.

HW for Thursday: Write up the inferences (hypothesis test and confidence interval) you generated today using the fluffy pom poms. Make your answers complete and correct. First period, please be prepared to meet me in our temporary location after our brief homeroom first thing Thursday.

HW due Wednesday: Complete write-up (PHANTOMS) of the Skittles hypothesis you generated in class. (Run a 1-proportion z-test on a large-enough sample of Skittles. The size of the sample depends on what you consider a "success.")

HW due Tuesday, March 18th: 12.13, 12.15, 12.16, 12.17. Please note: The answers in the back of the book for problem 13 used 0.33 as the value for 1/3. Obviously, this is not going to give you the "correct" answer. Please use 1/3 instead of their estimate and realize that your answers will be a little different from theirs, but yours will be correct! You can check your answers using the STAT TEST 1-PROP Z TEST on the calculator. The observation based on your results will not change just because you used a slightly different input.

The test on Chapter 12 will be next Tuesday, the 25th.

We're finally back where we began, with inferences for proportions. You should complete your compare/contrast study guide (that you created from scratch) where you compare the formulas, language, and assumptions of each of the three chapters. The headings are z-procedures for means, t-procedures for means, and z-procedures for proportions. Know how to justify each of the formulas for standard deviations in the two-proportion inferences algebraically.

HW: Using the combined data from two sleeves of mini M&Ms (as you will find in the holiday section of the grocery store), test the null hypothesis P = 1/6, where p represents the true proportion of brown M&Ms (the best ones!). Perform the complete test, using PHANTOMS as a guide. For those who were absent, we're talking about the little tiny plastic sleeves of M&Ms that come in a blue bag (about 17 to the bag). There are around 50 M&Ms in each sleeve unless you get short-changed.

Chapter 11 Statistical Inference Part II (t-procedures)

2008-03-10T15:29:00.001-04:00

HW for Wednesday, March 12: Find 10 problems from chapter 11 that have not been assigned yet and complete them. If you find problems that you cannot do, mark those, but they do not count toward your ten. YOUR TEST IS THURSDAY.

HW due Tuesday, March 11: 11.31 and 11.32. Keep the data from today's class in your calculator so we can finish the process tomorrow. Remember, the test is Thursday.

HW due Thursday, March 6: 11.10, 11.15. 11.17

HW due Friday, March 7 (Coach Rinehimer's birthday!!!): 11.16. 11.19, 11.20, 11.27, 11.30 (These are all related!) Good job with the pasta and grip labs today. My hypotheses were that there were 104 pieces of pasta on average in a cup and that the difference between right and left hand squeeze strengths was 7 pounds on average. Your job was to (1) collect data, (2) perform a hypothesis test using your results, and (3) construct a confidence interval for the average or averge difference. EACH of you is responsible for knowing how to do all the steps of these investigations.

HW due Monday, March 10: 11.26, 11.28

Your test on Chapter 11 will be Thursday, March 13th.

Happy Centennial!!!

W.S. Gosset's paper (remember, the brewer who developed the t-distribution?)was published in Biometrika 100 years ago THIS MONTH!
A link to the paper (of course he did not publish online. . .).

Chapter 10 Statistical Inference Part 1

2008-02-23T18:20:00.003-05:00

HW due Thursday: Problems 10.67 and 10.68 PLUS create a study guide for Chapter 10. YOUR TEST will be Tuesday, March 4. This is a short day, so you will have to think fast! HW due Monday: 10.70, 10.71, 10.78, 10.81, 10.82, 10.86. In addition to reviewing, we will be taking an ASMA test and starting Chapter 11 Thursday, Friday, and Monday.

HW due Wednesday: Problems 39, 40, and 43 done completely. Answers from the calculator are not sufficient.

HW due Tuesday, Feb 26: Problems 27, 28, 33, and 34 covering hypothesis testing. You may need to refer to the PHANTOMS side of the worksheet. If you have difficulty writing hypotheses, do problems 30, 31, and 32 as well.

RE: Hypotheses writing
The hypotheses always include PARAMETERS, not statistics. Use mu and p, not x-bar and p-hat. The null hypothesis always has an "equal to" nature rather than a greater than or less than.

HW due Monday, Feb 25: Problems from Chapter 10 (5, 12, 13, 14, 15). You will need to read the subsection, example, and info box before problem 13 to do the last few problems.

The AP registration deadline is upon us! Go to the Lassiter homepage and register for your exams. Bring a check to Ms Gasaway tomorrow.

HW for Friday, Feb 22: Do problems from Chapter 10: 10.6, 10.7, and 10.8 using your choice of SCAD or PHANTOMS and PANIC. The answers should be complete!

HW due Thursday, Feb 21: RE-DO problems 12.6-12.9 using the PHANTOMS or PANIC guides handed out in class. For those who miss Thursday's class due to a field trip, bring all homework on the day you return to demonstrate that you are current in the class or I will assign a more comprehensive set of problems upon your return to catch you up.

For Friday, Feb 22: continue to create confidence intervals, this time using x-bar and sigma of x. Do problems from Chapter 10: 10.6, 10.7, and 10.8. What assumptions or conditions do you need to check? If you guessed (1) SRS and independent observations, (2) population distribution is approximately normally distributed (or your sample is large enough for the central limit theorem to apply), and (3) the sample is less than one-tenth of the population size, then your brain is a finely-tuned instrument.

As you recall from Friday, the formula for the confidence interval for one proportion is p-hat +/- z* times SQRT(p-hat*(1-p-hat)/n)).

The source of the handout in today's class is
http://www.district196.org/evhs/People/baileyrcweb/APS%20Files/APS%20Main.htm

Please strike through the references to "retaining" the null hypothesis as well as the graph of a normal distribution on the confidence interval page. These are not appropriate for AP Statistics.

HW due Wednesday, Feb 20: 12.8 and 12.9 with complete solutions.

HW due Friday: Journal entry using problem 3 from the 2006 AP exam PLUS problems 12.6 and 12.7 from the text.

Alpha = the probability of rejecting the null hypothesis when the null hypothesis is true. It is the probability of a Type 1 error. This is the area associated with the rejection region.

HW due Thursday: Finish the 4-page worksheet about Distracted Drivers. Work on your journal entry (due Friday in a composition notebook).

HW due Tuesday: Problems 12.4 and 12.5. You'll have to read the information on the two pages preceding these questions. Come to class prepared to (1) explain how likely/unlikely your sample of pom-poms was from today's activity. That means that you have to calculate the z-value for your observation and compute the probability in the tail. ALSO, (2) be ready to explain how to simulate the distracted driver scenario using cards.

HW due Monday: Page 7 of 2006 Form B, problem 3 (golf balls) AND figure out a way to simulate the situation for problem 5 on page 9 of 2007 (the cell phone distraction problem).

JOURNAL ENTRY: The complete, perfect write-up of problem 3 from 2006 (page 8) will be due on Friday. The problem should be the first part of the entry, your fabulous answer should follow. This is the first problem to be entered in your composition notebook.

Has anyone tried this website? http://stattrek.com/AP-Statistics-1/AP-Statistics-Intro.aspx?Tutorial=ap

I wondered if it was helpful. Please let me know.

Chapter 9 Sampling Distributions

2008-02-04T21:30:00.000-05:00

Welcome to the beginning of inferential statistics! Your test is Thursday February 7.

Elaborate response to a question about mu and x-bar is in the comments section below. Don't miss it.

Work all the problems on the latest problem worksheet for homework Tuesday.

Mean of sampling distribution = mean of population regardless of shape of distribution

Std dev of sampling distribution = std dev of population divided by sqrt (n) regardless of shape of distribution as long as sample does not exceed 1/10th population.

Central Limit Theorem

Case 0: Underlying distribution is normal à sampling distribution is AUTOMATICALLY normal. The Central Limit Theorem DOES NOT APPLY. It isn't needed.

Case 1: If the sample is tiny (less than or equal to 10), then the population distribution must be nearly normal for the Central Limit Theorem to kick in.

Case 2: If the sample is moderate (up to about 35), then the population needs to be mound shaped without outliers for the Central Limit Theorem to kick in.

Case 3: If the sample is large, then the Central Limit Theorem kicks in.

****Write the implications of the Central Limit Theorem in your own words.

Thus, the answer to the questions at the end of problems .27 and .28 from Friday are that we DID NOT need the underlying distribution to be normal for us to use the formulas for the mean and the std dev of the sampling distribution (mu x = mu x-bar and std dev x-bar = std dev x / sqrt n, as shown in the text). Knowing that the distribution of x is Normal or knowing that the CLT applies is essential to working the sampling distribution problems using Normal methods.

HW: Problems 17 through 19 on the w/s from last week AND problems 8.34-8.36 on the w/s we handed out in class today.

Here's an applet that will help to show what happens to the distribution of the sample means as the sample size increases and as the distribution of the population is more or less normal. Set the radio buttons to show only the sample means and sample 100 at a time.
http://wise.cgu.edu/sdmmod/sdm_applet.asp

As you saw in the histograms today, the larger the sample size, the narrower the distribution of sample means. In fact, the standard deviation of the distribution of sample means shrinks proportionally to 1/( square root of the sample size)[the std dev of the sampling means = sigma of x times 1/sqrt n]. You'll see this in your READING of section 9.3 and practice it in your HOMEWORK for the weekend: 9.26 through 9.29.

For those who need to refresh their understanding of histograms, do problems 1.4 and 1.41 by hand.

Don't forget about CiCi's on Sunday if you are available.

What is the probability. . . ?http://www.time.com/time/health/article/0,8599,1707541,00.html?cnn=yes

Due Friday: Work problems 12-16 on the handout. Be an expert on these proportion problems and normal approximation for the binomial before you come to class. Friday we start sampling distributions for sample means.

Due Thursday: Problems 8-11 on the handout.

Due Wednesday: Problems 9.17 and 9.18, worked and explained completely, OR problems 9.19 and 9.20, which are more routine but must be completed.

What happens when the sample size increases??? If your variable of interest is the sample proportion, the distribution of sample proportions (p-hats) will become "tighter," that is that it will have less variability as the sample size increases.

Now, if you're talking about the values of X in a binomial distribution, as the number of trials increases, the variability of the number of successes also increases. Arrrrrggggh! Sometimes the standard deviation increases, sometimes it decreases.

One of the important ideas that you were supposed to catch was that the population size does not affect the variability of the sampling distribution. You "see" this when you look at the formulas--there is no mention of the population size in the formula for the standard deviation.

So, what is the point? For those binomial distributions where the expected number of successes and the expected number of failures are both at least ten, the sampling distribution of the Xs or the p-hats may be modeled (approximately) by the normal curve. Thus, you can calculate z-values for the values of interest, X or p-hat, and use the standard normal table or normalcdf to calculate probabilities. Cool.

This is part of the foundation for the polling estimates that you see so often during these election years.

Here's a new section for the blog called Because You Asked. One of the gurus of AP Statistics wrote an article that explains the 10% rule. You can find it through College Board at the following link.
http://apcentral.collegeboard.com/apc/members/courses/teachers_corner/39161.html

Due Tuesday: Problems 9.14 and 9.15 PLUS you must read the pages between these problems. Get your journal ready for the first entry.

Due Monday: All of the work previously assigned and a complete, correct draft of the answers to the "depth of the refracting layer" problem we worked in class on Thursday and Friday.

HW due Friday: Read through page 467 CAREFULLY. Work problems 9.1-9.4, 9.6, 9.8, 9.9, and 9.12. Do problem 9.7 if you get the chance (otherwise, it will be due later).

Parameters are measures of populations.
Statistics are measures of samples.

Examples of parameters: the mean of a population (mu, AKA mu sub x), the population proportion (p), the population standard deviation (sigma).

Examples of statistics: the sample mean (x-bar), the sample proportion (p-hat), and the sample standard deviation (s sub x).

The mean of all the sample means of a distribution (the sampling distribution) is the same as the mean of the distribution. This means that mu sub x-bar equals mu sub x.

The variability of the sample means (the variance of the sampling distribution)decreases as the sample size increases.

As long as the population is REALLY large compared to the sample, the size of the population does not affect the variability of the sample means.

HW for Thursday: Complete the blue sheet AND do problems 9.1-9.4 and 9.8.

Questions to ponder: What does the histogram of your penny-ages look like?

Does the histogram from a small sample have the same shape as a large sample's histogram?

What do you think that the average age of the pennies is?

How far out are outliers?

What are the important characteristics that you need to include when describing a probability histogram (either frequency diagram or relative frequency diagram)?

Chapter 8 Some special probability distributions

2008-01-16T11:20:00.000-05:00

TEST Tuesday 1/22/08
HW due 1/18: 8.41, 8.42
HW due 1/17: 8.37, 8.38, 8.39
HW due 1/16: 8.27, 8.28, 8.30, 8.31

HW due 1/15: 8.24, 8.25
With apologies to our Harvard buddies, here is a summary of binomial distribution facts and methods:
http://www.stat.yale.edu/Courses/1997-98/101/binom.htm

HW due 1/14: 8.19, .20, .22, .23

You guys rocked today! Your questions and explanations were top notch. HW 1/10 Answer problem #11 fromt he worksheet in the blue book using all three of the methods: formula, binomial pdf, and binomial cdf. Show all work and include explanations that you can rely upon to remind you of these methods later in the semester. Be neat and write legibly. Use appropriate language.

HW 1/9 Answer the questions from the 12 days worksheet that have a binomial setting.

HW 1/8 Write up a study guide for the Binomial Probability section of Chapter 8 AND answer questions 8.1-8.4. Bring the "12 days" worksheet back so we can discuss it.

Check out the arbitrary (not random) stuff in the next blog!

First semester final exam and other neat stuff to read

2008-01-06T14:00:00.000-05:00

A timely article about polling: http://www.cnn.com/2008/POLITICS/01/02/projection.explainer/index.html?iref=newssearch

Article on the effect of improved organizational skills on boys' achievement in school:
http://www.iht.com/articles/2008/01/01/america/01boys.php

Have you seen the Norwegian engineering recruitment videos yet? Scroll down and let me know what you think.

For info on influential points and outliers, google statistics influential points. Some students posted pretty good (but terribly distracting) powerpoints.

The SAT results for Georgia broken down in all kinds of interesting ways:
http://www.collegeboard.com/prod_downloads/about/news_info/cbsenior/yr2007/GA_07.pdf
These data prompt discussion about causation, such as whether choosing a technical major makes your math score higher. What do you think?

On the lack of publicity for Nobel Prize Winners:
http://pharmtech.findpharma.com/pharmtech/From+the+Editor/What-Good-is-Winning-a-Prize-if-No-One-Cares/ArticleStandard/Article/detail/469658

So you want to be a casino mogul? How about starting with one of these jobs?
http://www.state.nj.us/oag/ge/2000news/slotlab.htm

Shortages of engineers (Click on the READ online button, then click on the lower right corner of the pages to advance through the newsletter)
http://thefloridaengineer.eng.ufl.edu/themagazine/fe0801/index.php

Two videos from Norway:

There's one bad word in this one written in the subtitles. Please excuse this word--the rest of the video promotes engineering for young people. (And you know how those crazy Norwegians are.)
http://www.youtube.com/watch?v=gXZZknJWtGg&feature=related

http://www.youtube.com/watch?v=JFoMmJAAHjg&feature=related

Water use statistics for Cobb: http://www.ajc.com/metro/content/metro/cobb/stories/2007/12/19/cobb_water_1220.html

Interesting webpage about one of our community members and how his company provides service in disaster relief:
http://cateringcajun.net/_wsn/page7.html

Chance, conditional probability, and MARS!
http://www.cnn.com/2007/TECH/space/12/21/mars.asteroid.ap/index.html

Chapter 7 Random Variables

2007-12-11T15:20:00.000-05:00

INTERESTING NEWS:
http://news.yahoo.com/s/ap/20071213/ap_on_re_us/hiv_lawsuit

This test will be on Thursday, December 13, 2007.

You will need to create a chart to remind yourself about the formulas for this chapter until you have practiced enough to know them by heart. A sticky-note at pages 396 and 400 would also be helpful!

Mu = the POPULATION average. This is a parameter.
Sigma squared = the POPULATION variance. This is also a parameter.
Standard deviation is the square root of the variance.

X-bar is the sample average, the unbiased estimator of the population average. It is a statistic.
S-squared is the sample variance, the unbiased estimator of the population variance. It is also a statistic.

HW due Wednesday, December 12th: 7.13, 7.15, 7.17, 7.34, 7.42
HW due Tuesday, December 11th: 7.24, 7.28. Use the formulas and the examples in the text.

HW due Monday, December 10th : 7.2, 7.4, 7.7.

Chapter 6 Probability

2007-12-03T15:10:00.000-05:00

The Chapter 6 test will be December 6th.
Previous tests will be returned to students as soon as they are graded.

Prepare for the test. Work problems from each section. Read the chapter and section summaries. Write down what you are doing. Draw the Venn diagram or the tree diagram for complicated situations. Ask questions on the blog. Take the practice test.

Here are some answers to even HW problems:
Problem 6.10 (a) S= {all numbers between 0 and 24}
(b) = {any whole number up to and including 11,000}
(c) S = {0, 1, . . . 12}
(d) S= {any dollar and cents amount up to [insert your maximum guess here]}
(e) S = {any positive or negative number}

Problem 6.12 Four outcomes for two coins: {HH, HT, TH, TT}, eight for three coins: {HHH. HHT, HTH, HTT, THH, THT, TTH, TTT}, and sixteen for four coins (do that one yourself!).

Problem 6.16 (a) YYY -0000 through YYY-9999 = 10,000 numbers. (b) YYY-ZXX-XXXX, each X having 10 possible numbers, except the number can't start with a 0 or a 1 meaning that Z has only 8 possible values, so this means 8 * 10^6 LESS the restricted numbers (911-xxxx, 411-xxxx, etc.)

Problem 6.20 P(moves to another class) = 1 - P(stays) = 1 - .46 = .54.

Problem 6.24 P(wins large battle) = .6, P(wins three small battles) = P(wins individual small battle)^3 = .8^3 = .512. Choose the strategy with the larger probability of occurring.

Problem 6.40
Venn diagram has two circles representing getting job A and getting job B.
Both jobs: intersection of the two circles, the overlapped part, the biscuit.
First but not second: the part of circle A that is not within circle B
Second but not first: the part of circle B that is not within circle A
Neither: the part that is in the background, in NEITHER circle.

Problem 6.44
P(W) = 856/1626
P(W given prof degree) = 30/74
These are not the same. so gender and professional degree are not independent

Problem 6.56
P(y <> x) = 1/8,
P(y > x) = 1/2,
P(y <> x) = P(y <> x) /P(y > x) = 1/4.

Problem 6.48: P(W) * P(Manager given W) = P(Woman AND Manager)

One pattern that shows up a lot is Marginal * Conditional = Joint

If you divide both sides by Marginal you get
Conditional = Joint / Marginal.

IFF means IF AND ONLY IF.

IFF P(A) * P(B) = P(both A&B), A and B are independent.

IFF P(A) = P(A given B), A and B are independent.

HW for Tuesday night: DO problems 6.33 and 6.48. Read 6.66 and be prepared to work the problem. Essential question: How do mathematical independence and our regular understanding of independence relate? The chapter 4 tests were returned today. HW for Monday night: 6.39, .40, .53, and .56. The problem we worked today in class was problem .65. You would be wise to work through this problem and problem .66.

Notes from Friday (11/30) are embedded in the purple sections below.

Conditional probability rules:

PLEASE NOTE THE CORRECTION! BLOGGER WON"T ACCEPT THE VERTICAL LINE SYMBOL!!!
P(A GIVEN B) = P(A and B)/P(B)
P(B GIVEN A) = P(A and B)/P(A)

so of course

P(B) P(A GIVEN B) = P(A and B), the joint probability of A and B. It may be helpful to think of it like cancelling factors in the numerator and denominator of a fraction EXCEPT that the result is the JOINT probability. Be careful.

P(A) P(B GIVEN A) = P(A and B), again, the joint probability of A and B.

These relationships can be represented in two-way tables, Venn diagrams, and tree diagrams. The count within a cell of a two-way table divided by the marginal total is a conditional probability. Likewise, the joint probability for that cell divided by the marginal probability is also the conditional probability.

Tree diagrams can be useful when you are trying to work the problems backwards.

I don't think that I made this clear in class today:
P(A) = P(A and B) + P(A and not B) = P(A)P(B given A) + P(not A) P(B given not A).

HW for the weekend is problems 6.44 and 45.

If A and B are independent, then P(A) = P(A GIVEN B) and P(B) = P(B GIVEN A).
Interpretation: If A and B are independent, then whether or not B happened has no relationship with whether A happened.
Likewise, if A and B are independent, then whether or not A happened has no relationship with whether B happened

Today we used a Venn Diagram, a two-way table, and a tree diagram to represent the outcomes and probabilities associated with throwing two strangely-marked dice. All of the methods yielded the same answer.

HW for Tuesday (11/27) night: Re-work the weird dice problem from the AP exam (the one with two dice, one has only 9s and 0s, the other has 11s and 3s.) This time, instead of using simulation, use formal probability rules and a tree diagram, table, or Venn diagram. Answer the question in complete sentences. For part B, reconcile the answer with the joint probabilities you found in part A. Figure out the guidelines in your own words that tell you whether a price/reward is fair.

Get the reading done! What are the big concepts?
----------------------------------------------------------

Sorry for the delay: just got home from KSU.
HW for Monday night, Nov 26: 6.19, 20, 21
plus. . .finish State of Fear.
-----------------------------------------------------------
Add problems 6.24 and 6.25, due Monday, November 26.

Don't forget to read State of Fear.
------------------------------------------------------------
he complete list of HW problems due Tuesday: 6.9, 10, 12, 13, 16 (plus any others you feel like doing).

Don't forget to read State of Fear.

------------------------------------------------------------

Events that are mutually exclusive ARE NOT independent.

Addition principle: P(A or B) = P(A) + P(B) - P(A and B)

Multiplication principle: P(A) P(BA) = P(A and B), the joint probability.

When B and A are independent, P(BA) = P(B), A happening or not has not relationship to B happening, so P(A) P(B)=P(A and B). THAT IS ONLY WHEN THE EVENTS ARE INDEPENDENT.

Key vocabulary
parameter
sample space
event
probability
joint probability
independent
---------------------------------------------------------------------------
This won't be so bad. The test will be Thursday, Dec. 6.

What are YOU doing to maximize your understanding of the material?

Are you creating an outline of the chapter?
Have you developed a glossary for the vocabulary and formulas?
Have you worked all of the homework problems when assigned?
Do you read the sections that relate to the homework?
Are you part of a study group?
Do you ask questions?
Have you worked problems from a study guide?
Have you worked the online quiz (see the link on the right panel of this blog)?

Do you try to see the big picture?

Do you look for the similarities and differences in the ways data are processed?
Do you work with problems long enough to understand why the formulas work the way they do?
Have you made connections between current concepts and prior knowledge?
Have you gone online to review concepts that you have forgotten?

Just "going through the motions" does not lead to the success that you desire in Advanced Placement courses. Take control of your learning.

Be safe.

Chapter 5 -- Producing Data

2007-11-14T16:00:00.000-05:00

Be sure to check the comments for this post. Your fellow classmates ask the best questions! Have you taken the online quiz? Have you worked through a study guide? Have you prepared for Thursday's test? Have you finished the reading for Tuesday? Why would I ask you to read that book in the middle of Chapter 5???? [There must be a reason.] Why would you use simulation instead of actually testing the real thing in an experiment? What are the three essential principles of good experimental design? Why is each one important? What does bias have to do with all of this? What IS bias? What IS confounding? When do you block? What is the difference between stratifying and blocking?

The two sweet diagrams of experimental design are on page 272 (Completely randomized) and page 280 (block design).

Blocking is a form of control. When a large number of your experimental units share some pre-existing condition that may make their responses to the treatment vary tremendously WITHIN the treatment groups, you will have a hard time differentiating between the results of the treatment groups. You would really prefer to have the differences in results BETWEEN groups to be big enough so you can make a decision about your comparison. To reduce this vaiability, you may choose to BLOCK by the nuisance variable (the pre-existing condition). Then you RANDOMLY allocate the experimental units in each block to the different treatments. If there are two treatments, then each block is randomly broken into two treatment groups. You proceed by running the experiment on each of the blocks individually.

Thursday and Friday (11/8-11/9) Finish writing up the experiment described below, using all the concepts of section 5.2. ALSO, answer the free response (FR) problems from 2001 and 2003 handed out in class. You definitely NEED to read the section of the book. These are NOT opinion problems.

The Chapter 5 test is Thursday, November 15. Freakonomics must be read by 11/13.

Wednesday night's (11/7) HW: 5.65 PLUS design an experiment to answer this question (at least 6-7 sentences!!).

Does the choice of presentation technology make a difference in student achievement in a geometry class?

Conditions: Geometry classes at Lassiter
four teachers teach geometry
some teachers have students write HW on the board
some teachers have students write HW answers on the overhead projector
some teachers put their official answer transparencies on the O/H.

Two document cameras are available to use (Google document camera if you haven't seen one!)

Students are already assigned to the classes.

How could we design this experiment to answer the question? What questions or clarifications do you have? Bring at least seven complete sentences of helpful guidelines for performing this study.

Friday night's HW: 5.63 and 5.64. Complete most of your Freakonomics assignment this weekend. When you get to the part where the authors belabor their unique name theory, you can consider your assignment completed. what was your favorite part? What connections did the authors make that you agree with? that you don't agree with?

Thursday night's HW: 5.60 and 5.61
Wednesday night's HW: 5.54, 5.55, 5.56 Be safe.

Tuesday night's HW: Complete both of the problems from the 2001 exam.

Example of using the TORD to simulate a bag of M&Ms with the OLD color distribution:

Old Distribution:
Brown 30%
Red 20%
Yellow 20%
Green 10%
Blue 10%
Orange 10%

Let's try this two ways. First, let's use two-digit numbers to simulate candies according to the following schedule.
01-30 Brown
31-50 Red
51-70 Yellow
71-80 Green
81-90 Blue
91-00 Orange

There are no excluded numbers. If we draw the same number twice, use it again!

Using the following line from a table of random digits, simulate drawing 5 candies.

63996 32914

63>>>>Yellow
99>>>>Orange
63>>>>Yellow
29>>>>Brown
14>>>>Brown

The second way requires only one digit. Let 1-3 represent Brown, 4-5 for Red, 6-7 for Yellow, 8 for Green, 9 for Blue, and 0 for Orange.

21833 70905
Using the TORD above, you would get
2 Brown
1 Brown
8 Green
3 Brown
3 Brown

Link to interesting site about the Dewey-Truman polling error. Did you know who the third party candidate was who threw the wrench into the process? Strom Thurmond. Your parents will be impressed that you know this.

http://www.hannibal.net/stories/101998/Pollstersrecall.html

Interesting historical link about Tukey. Scroll to the middle to see his influence in predicting outcomes of elections.

http://www.amstat.org/about/statisticians/index.cfm?fuseaction=biosinfo&BioID=14

The two books I assigned for November are Freakonomics and State of Fear. Freakonomics discusses a lot of associations/correlations that promote critical thinking. State of Fear makes you enlightened consumers of research (even though it IS fiction). Many parents have probably already read one or both of these books. Last year's students (generally) loved them.

Wednesday, October 24
Take notes on the first section of the new chapter, especially new vocabulary.

For Monday and Tuesday of next week:

5.1-5.5, 5.8, 5.11, 5.17-5.18, 5.22, 5.23

Key concepts covered in class (alliteration, anyone??) today included

undercoverage
non-response bias
response bias
convenience sampling
voluntary response sample

and examples like the C-SPAN and American Idol calls, surveying the people sitting around you, the Dewey Defeats Truman mistake, answering with un-truths, failure to respond to surveys.

Can you match the concept to the example? Can you think of another example of each concept in action? Why does each of these result in data we cannot rely on?

Non-linear relationships

2007-10-08T14:42:00.000-04:00

Assignment for Monday: Create an outline of the key points in the chapter, including all vocabulary words.

Also, do problems 4.54, 4.56, 4.62 (this study just celebrated its 20th anniversary!!!), and 4.66

So, how about those marginal and conditional distributions for two-way tables, huh?

The marginal distributions are the percents that each column or row represents in the entire table. For instance, if the total of one row was 250 and the total for the table was 1000, the marginal distribution for that row is 25%. You would continue to calculate percents for all of the other rows or the other columns--whatever the question asked for.

For the conditional distributions, you only consider a portion of your population, for instance only a specific row or column. Then, what portion of the observations recorded in tht small group shared the desired characteristic?

If there were 15 sophomores taking AP Chinese and 500 sophomores in a school of 2000 students, GIVEN THAT a student is a sophomore, the percent who are taking AP Chinese is 100*15/500.

Tuesday. October 16

Assignment for Friday: 4.34, .36, .38, .39, .40, .42, and .43

How did you like the Simpson's Paradox activity today?

When breaking data into two or more divisions by a lurking variable changes the "decision" for EVERY ONE of the sub-groups, the result is a Simpson's Paradox. For instance, the example today presented no clear, justifiable answer about whether we should fund Bolgg's Panacea or not.

The example in the book about the hospitals is instructive.

Good luck on the PSAT.

Monday, October 15

4.22-4.24, 4.27, and 4.28

Review the topics and procedures on the notes handed out today.

Have you ever heard of Simpson's Paradox???

Thursday, October 11
We've transformed non-linear data to a linear form, found the LSRL through the data, re-written the equation reflecting the nature of the lists used to develop the LSRL, and re-transformed the equation to model the original data.

You should have done problems 4.6 and 4.9. For tonight, DO problem 13 and READ ACTIVITY 4 and problem 4.15. If you feel excited about the investigation, read problem 16 also.

Monday, October 8

We graphed some relationships between x and y to determine whether we were allowed to run the LSRL on the data. Of course, we ONLY run the least squares regression on data that look like they have a linear pattern.

When the pattern in L1 and L2 looked like an exponential growth or decay model, we took the log of y in order to un-do the exponential. Putting the log y into L3, we proceeded to verify that the graph of L1 and L3 was approximately linear. We then ran the LSRL through that set of points.

The equation we found by using the LSRL will not run through our curve-y data, so we have to un-transform the equation. For the exponential case, we had used the log of y instead of y itself when finding the LSRL (but the original x values!), so we re-write the equation as log y-hat = a + bx.

We solve for y by taking the antilog of both sides ("ten-to-the" or 10^stuff). The resulting equation for y can be graphed with the original x and y data and should match the pattern pretty well.

If the model looks like a quadratic, square root, or other power function, you'll need to perform mostly the same functions, but on the logs of both x and y. The linear equation that passes through the straightened data will be transformed like this: log y-hat = a + b times log x, and the result will have a factor equal to x to the b power.

HW: Problem 4.1. The answer is in the back of the book, but there are a lot of sections to this problem.

Chapter 3 Linear Relationships

2007-09-20T13:13:00.000-04:00

Tuesday, October 2
Work AT LEAST 5 problems to prepare for Thursday's test.

Monday, October 1
3.50 and 3.52

Test on Thursday

Friday,September 28
Yo, this is David T.

Well today we broke up the deviation in a prediction, part into y-hat minus y-bar and an error part y -y-hat.

We also explained the significance of r^2 which equals the portion of the variation in y which could have been predicted using the regression relation.

Remember that if r^2 is close to zero, then the points on the graph are crazy and scattered. If r^2 is close to one, then the graph and points are predictable and are linear.

The HW is 3.46 and 3.49. (YES, this means YOU!) You = David V.

Thursday, September 27
KEY FORMULAS

b = r * Sy/Sx
a = y-bar minus b* x-bar

y-hat = a + b* x

residual = actual minus expected = y - y-hat

If residuals are small and scattered, then the linear model is a good model. If there is a distinct pattern (if you could predict what the residual would be for a particular x-value), then the linear model is not appropriate.

Be sure to WRITE what you see in the residuals ("The residuals are small and scattered, so a linear model is appropriate" or not) and what effect that observation has on your model.

Also be sure to write out the description of the y-hat equation in words: "The predicted value of [insert y variable here] is approximately [insert y-intercept here] plus [insert slope here] times [insert the x variable here]."

COMMON ERRORS:
Failure to use LinReg(a+bx) L1, L2, Y1

Failure to check that the observed y values are close to the predicted y values.

Failure to use the same x and y in your stat plot that you used in your linreg equation. (causes graphs to not show up!)

HW problem 3.39.

Wednesday, September 26
Problems 23 and 31 PLUS find the Least squares regression line for the Archaeopteryx data.

Tuesday, September 25
We re-worked the HW from last night and extended the concept by investigating what happens when you calculate the correlation coefficient for non-linear data (Anarchy! Riots! Dogs and cats living together!). Although you CAN calculate a correlation coefficient for non-linear data, the results tells you NOTHING.

Key points to remember:

-1<= r <= 1. Always. No getting around it.
r is dimensionless. If you change units or perform a linear operation on all of the values of x, or y, or both, your r will not change!!! In fact, what happens when you switch the order of the variables and calculate r for L2 and L1????
r is affected by outliers. They increase the standard deviation, which causes the denominator to be smaller, which causes the r to be closer to 0.
r only gives you information about linear relationships. If it isn't linear, then this linear modeling is inappropriate.

If you haven't already tried it, calculate r for some small sets of non-linear data and see what I mean.

HW 3.13 and 3.19. All about the archaeopteryx.

See you tomorrow.

Monday, September 24
Do problem 3.18. This is just like what we did in class.

Friday, September 21
Problems 3.1-3.3 and 3.5.

Fifth and sixth periods: You did a great job with all the distractions today. Thanks for trying to stay on task.

Good job, Trojans! You make us proud.

CiCi's on Sunday? 2-4.

Be safe.

A new Chapter!!!

Thursday, September 20

Copy the formulas and definitions from Chapter 3 into your notes.

Chapter 2 Probability Distributions

2007-09-07T16:20:00.000-04:00

Summaries of current topics can be found below the homework details.

Tuesday, 9/18
Work three of the problems that you set up last night and in class today. ALSO, do problems 2.41, .42, and .43 completely.

What more do you need in order to be successful with this? Make a list! Outline the important concepts from the chapter!

Why is the normal distribution such a big deal?

Monday, 9/17
Select 15 problems from Chapter 2. Split your paper in half (hotdog style). Write the details of the GIVENS on the left and the items that the problem asks you to find on the right side. You do not have to solve the problems. Watch carefully for those cases where there are multiple requirements.

Your test is Thursday.

Wednesday, 9/12

UPDATE: I have posted a WORD document with hints on the homework site: classhomework.com. You'll have to enter the password and then click on the file name.

Post to the blog and let me know when you get it--but don't ruin the fun for the other students!
-------------------------------------------------------------------------------------------
You KNOW that the area under the curve in a probability distribution is always one (That's why you get 1 when you use normalcdf(-infinity, +infinity)). Go back to the first parts of Chapter 2 and review the characteristics of a probability distribution.

Then, for HW, find the values of x which represent the Q1, median, and Q3 of a triangular probability distribution that starts at the origin and ends at (4, ???). Yes, you have to figure out what the value of ??? is so the are under the curve is 1.

You will use the formula for the area of a triangle: A = (1/2) base * height.

There are many different ways of attacking this problem. How many can you find???

Tuesday 9/11
HW 2.24, 2.25, 2.30

If you don't understand something, ask a question on the blog. Coming to class unprepared is not an option.

Monday 9/10
HW 2.15, 2.16, 2.22, 2.23

Friday 9/7
You developed equations today to standardize observed values.

Your formula for the z-score was (observed x minus the average)/(standard deviation).

You also used a formula to find the value of x that has a certain z-value:
x = average value + (z-score)(standard deviation)

but you probably noticed that the second formula is redundant.

You also learned about the Empirical Rule.
http://www.stat.tamu.edu/~west/applets/empiricalrule.html

This ONLY WORKS with normal distributions and it is only an approximation. We will learn more precise methods next week.

If you're interested in a neat relationship that works for other types of distributions, check this out: http://www.stat.tamu.edu/stat30x/notes/node33.html

Another neat website:
http://people.hofstra.edu/stefan_waner/Realworld/Summary7.html

Now, is anyone out there planning to CiCi's this Sunday? I won't go unless there is interest, so post your plans!

HW: 2.6, 2.7, 2.8 and read up to that point in Chapter 2. Be safe.

Graphs and standard deviation & CiCi's

2007-08-29T17:50:00.000-04:00

Preparing for the test
Your test is Thursday. Begin to prepare now by working problems and creating an outline.

Homework: For those in class today - rewrite your responses to the FR questions, plus work problem 1.4 from the text.

For those absent today: Problem 1.4 PLUS ALL OF 1.48-1.52. Pick up your original responses to the FR upon your return to complete overnight. If you were participating in Senior Skip Day, your absence is unexcused.

CiCI's
YES!!!! There is a request for CiCi's this Sunday, so I will be there from 2 to 4. That is the one by the Walmart at Trickum and 92 (close to Arby's).

Test is Thursday, 9/6!

If you are going on the marketing fieldtrip, stay after school to take the test in room 214 at 3:30. Don't forget to bring your calculator.

For Tuesday (9/4), select one odd and one even problem from the set 1.48-1.52 and work them completely. Become an expert on one of the problems.

For Friday (8/31), complete problems 1.41 and 1.43. The answers are in the back of the book, but that is not sufficient! You must show all work and explain your actions.

By Thursday (8/30), you should have both the graph from the Internet, a newspaper, or magazine and problems 1.35 and 1.36 from the text.

RE: the graph
You will identify the variable(s) represented in the graph and the type of graph you brought. Are the data numerical or categorical? Are numerical data discrete or continuous? Does your graph represent one variables or two? Is a trendline appropriate for your data?

RE: Standard deviation

The standard deviation of a sample of data is like an average deviation from the sample mean. It is the square root of the sample variance, which is an unbiased estimator of the population variance.

If we just found the sum of the deviations, we would get a sum of zero because some data are above the mean and some are below. Because of the definition of the mean, the positives and the negatives cancel each other out.

Instead, we square each deviation so the numbers we add together are all positive. We "average" these numbers by dividing by (n-1). You remember that n is the number of observations. We subtract one because we are using an estimate derived from the data themselves for x-bar. This gives us the sample variance or s-squared. To get the value of s just take the square root.

In formula form, s = sqrt(sum of all the squared deviations/(n-1)). The formula for the first squared deviation is (x minus x-bar)^2. Again, x-bar is the average of the x values.

The same relationship holds between sigma and sigma squared, the population standard deviation and the population variance: you take the square root of the variance to get the standard deviation.

Cumulative and relative frequency histograms

2007-08-27T15:18:00.000-04:00

We created cumulative frequency histograms and relative frequency histograms today. For the cumulative frequency histograms, find the cumulative sum up to and including each line of the frequency table, for instance,

x freq cumulative freq
1-5 4 4
5-10 5 9
11-15 6 15

For the relative frequency distribution, divide the count for an interval by the total number of observations, n. What do you observe about the graphs of frequency and relative frequency????

The HW is attached to the site at classhomework.com.

Histograms - a beginning

2007-08-24T15:24:00.001-04:00

Find a nice set of data to graph.

Create a frequency table for the data, breaking the data into 5, 7, or 9 intervals of equal length. That does NOT mean that there will be an equal number of observations in each bin or interval!

Create a histogram to represent the data.

http://www.ncsu.edu/labwrite/res/gt/gt-bar-home.html in Excel

http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Estes/graphicaldisplays1page.html on the TI-83, but you need to reset the window!!!

http://facstaff.colstate.edu/henning_cindy/Calculator%20Assistance_files/Creating%20Histograms%20on%20TI83.htm

Stem and leaf plots

2007-08-23T15:18:00.000-04:00

Create a stem-and-leaf graph (a stemplot) of your data. If your data your data don't go nicely into a stemplot, find some fun data to use instead.

More later. . .

Have you found good resources on the web?

Do these help?
http://regentsprep.org/regents/math/data/stemleaf.htm

http://en.wikipedia.org/wiki/Stemplot

http://www.sjsu.edu/faculty/gerstman/StatPrimer/freq.pdf

More box and whisker

2007-08-22T11:19:00.000-04:00

Using the data from problem 1 of the 2001 exam, answer the questions posed in class. Part C reads, "The news media reported that in a particular year, there were only 10 inches of rainfall. Use the information provided to comment on this reported statement.

Keep in mind all the errors that students might have made under test conditions. What do you suppose that a student under extreme time pressure might have done wrong on this problem?

Box and whisker plots

2007-08-21T13:14:00.000-04:00

You collected data today in class and we talked through the processes of finding the five number summary and constructing a box and whisker plot for univariate quantitative data.

5 number summary: Min, Q1, Med, Q3, Max

Use the 5 number summary to construct the box and whisker plot. Use the interquartile range (the length of the box containing the middle 50% of the distribution)to determine whether observations are outliers. The boundaries of reasonable answers are Q1 - 1.5(IQR) and Q3 + 1.5(IQR).
**********************
For homework, construct the modified box and whisker plot for your data.
**********************
There are many websites that explain how to perform this task.
http://www.statcan.ca/english/edu/power/ch12/plots.htm

Can you find one that you like better? Please share the site with the rest of us.

Types of variables

2007-08-20T11:19:00.000-04:00

You further analyzed the data you collected last week and made a claim about whether we can use the x-value to predict the y-value for your experience.

We discussed many types of variables in class today, focusing on quantitative (discrete and continuous) and categorical.

Create a list of 20 clever variables and identify whether each is quantitative or categorical. Include both types in your list. If the variable is quantitative, determine whether it is discrete or continuous.

Examples:

The number of students in math classes at Lassiter: quantitative and discrete.
The number of minutes of studying/homework done by students each night: quantitative and continuous.
The math course taken by students at Lassiter: categorical. [The values that the variable can take are the different courses.]

The linear labs

2007-08-17T14:41:00.000-04:00

Find the least-squares regression line through the data you collected in class today. Describe trends in your data, the direction, linearity, strength of the relationship, and presence of outliers. Present a comment about the nature of the connection between the x and y values from your lab.

First period:

These may not be your actual data. . . the postits got a little mixed up. Anyway, you can use these for your HW.

Other classes:
If you lost your data, post a message asking for your particular set of data. NO LAST NAMES PLEASE!!!!

Arm/foot
12, 10.5
11, 10
11, 9.5
11, 10
11.5, 10
11, 10
11, 10
11.5, 10.25
12, 10.5
11.5, 10
12, 11.5
11.5, 10
11, 9
10, 8.5
9.5, 9.25
11, 9
8.5, 9
8.5, 9

Days
15, 16
1, 30
3, 28
9, 21
8, 22
12, 18
11, 17
14, 17
15, 16
17, 14
19, 12
7, 23
26, 5
24, 7
25, 6
24, 6
30, 1
30, 1

Ball toss
5, 6
3, 3
3, 3
2, 4
1, 2
2, 1
1, 0
0, 0

The ball-measurement lab

2007-08-16T11:48:00.000-04:00

You did a good job using the center, shape, and spread of data to match the graphs to the summary statistics for last night's assignment.

The data you collected will be posted here. Please post comments and questions attached to this entry. Our technical team is standing by, ready to answer your questions!

Your assignment is to generate a least-squares regression line for the data. Your technical team will post instructions.

Do not use your full name when you create your account.

Here are data you collected:

Circumference Diameter
25 3.5
24.75 3.5
19.5 2.5
19.875 3.25
19.75 3
19.75 2.5
20.5 3
16.5 2.25
16 2
15.25 2
12.25 1.5
15.25 2
8 1
8.5 1
12 1.5
12.5 1.75
8.25 1.25
12 1.75
13 1.75
5.5 0.75
3.5 0.5
5 0.5
3.5 0.5
5.5 0.75
4.75 0.5
5.5 0.75
9 1.25
12.5 1.75
15.25 2.25
20 3
19.75 2.5
25 3.5
9 1.25
12.2 1.75
8 1
5.5 0.75
4.75 0.75
8.25 1.25
20 5.5
33 8.5

Welcome to the new school year!

2007-08-14T13:17:00.000-04:00

You did a marvelous job of spinning pennies today. Can you think of ways to minimize the outside influences on the results of the spin? How could you make the results dependent only on the fairness of the coin--well, as much as possible?

Your homework can be found on Classhomework.com.

Some of our classmates have reported that the composition notebooks are sold out of local stores. Don't panic. We won't do our first write-up for at least another week.

Also, don't rush out to buy a study guide or a new calculator. The new edition of the Barron's guide will be released in September. TI will be releasing the new calculator, the TI-Inspire in September as well. There is no sense in spending money on last-year's model.

We will issue textbooks when we finally need them. We will do some more investigations (labs) in class before we use the text.

When you (eventually) need to retrieve a document from the HW website, the password that you will use is lassitermath.

Leave a comment if you want to share thoughts or questions.

The final weeks

2007-05-13T13:24:00.000-04:00

Clarification: The AP exam exemption policy says that you must be passing the class and have fewer than 6 absences to exempt the final exam. Anyone who is not passing must take the exam.

If you are part of the crowd who has to take the final, then take those practice exams that you used to prepare for the real exam. Your exam will be multiple choice.

Getting ready for the exam

2007-05-06T17:48:00.000-04:00

I have sent out emails to 3rd period teachers asking for you to be released in time to eat (except for JROTC and pers fitness. Please let them know!!!).

Meet at the outdoor classroom between 11 and 11:30.
Bring pencils, calculator, pen, smile, sweater, mittens, scarf, hand warmers. . .

Just kidding a little about the stuff to keep you warm. Reports from Monday's exams were that it was FREEZING in the gym. Bundle up in layers.

Some sites to visit:
for fun
http://www.youtube.com/watch?v=Ooa8nHKPZ5k
for real
http://tinyurl.com/gwcmq

Somethings to remember:
You CANNOT discuss the MC problems at all--not in person, not on the phone, not on the web.

You can discuss the free response problems after 4:00 on Thursday.

Bring pencils and a pen to the test. Do not bring your cell phones, i-pods, etc. You can leave them in my room. I will lock them up.

Know your assumptions/conditions.

What questions do you have?

Chapter 13 Inferences for regression

2007-04-30T15:51:00.000-04:00

Here are the answers from today's activities:
Please forgive formatting. Note that the values of SECoef, T, and P for the constant are not used in our inference calculations.

Regression Analysis: C6 versus C5
The regression equation is
C6 = 26.7 + 57.0 C5

Predictor Coef SECoef T P
Constant 26.75 19.44 1.38 0.263
C5 57.00 42.17 1.35 0.269

S = 9.42956 R-Sq = 37.8% R-Sq(adj) = 17.1%

Regression Analysis: C9 versus C8
The regression equation is
C9 = 32.4 + 37.5 C8

Predictor Coef SECoef T P
Constant 32.38 25.87 1.25 0.299
C8 37.50 60.19 0.62 0.577

S = 14.1877 R-Sq = 11.5% R-Sq(adj) = 0.0%

Regression Analysis: C12 versus C11
The regression equation is
C12 = 17.5 + 90.3 C11

Predictor Coef SECoef T P
Constant 17.50 10.45 1.68 0.192
C11 90.31 14.96 6.04 0.009

S = 8.52974 R-Sq = 92.4% R-Sq(adj) = 89.9%
--------------------------------------------
Regression Analysis: C15 versus C14
The regression equation is
C15 = 52.7 - 14.0 C14

Predictor Coef SECoef T P
Constant 52.67 18.74 2.81 0.067
C14 -14.00 42.50 -0.33 0.764

S = 7.76030 R-Sq = 3.5% R-Sq(adj) = 0.0%
-------------------------------------------

Descriptive Statistics: C19, C20
Variable N N* Mean SE Mean StDev
C19 5 0 0.5000 0.0791 0.1768
C20 5 0 61.00 8.34 18.64

Regression Analysis: C20 versus C19
The regression equation is
C20 = 9.00 + 104 C19

Predictor Coef SECoef T P
Constant 9.000 5.279 1.70 0.187
C19 104.00 10.07 10.33 0.002

S = 3.55903 R-Sq = 97.3% R-Sq(adj) = 96.4%

--------------------------------------------

Now, can you generate a confidence interval for the slope of the REAL regression line from one of your estimates? What does your (large) interval tell you about the strength of the relationship between x and y?

What were the three types of evidence you used to answer the questions about the model? Match the evidence to the question online at the quizplace. http://www.proprofs.com/quiz-school/quizview.php?id=968

Chapter 13 Inferences using Chi-square procedures

2007-04-13T16:54:00.000-04:00

Hang in there; we're getting close to the end of the race.

This chapter introduces us to chi-square procedures. These methods are generally used to analyze tables of counts from samples which are separated into CELLS based on one or more categorical variables. The advantage of these methods is that you can perform many comparisons at once, instead of just two as in our previous procedures using z and t. Most students like chi-square procedures better than z and t procedures because we will be using counts rather than continuous data and our tests are automatically two-tailed.

For instance, we might analyze the COUNTS of M&Ms of each of the six usual colors in a bag or the distribution of the COUNTS of teachers at each combination of YEARS OF EXPERIENCE and HIGHEST DEGREE ATTAINED. Each element counted must be placed in exactly one CELL. We will compare the OBSERVED counts from a sample or samples to the EXPECTED counts in a way that will quantify the likelihood of this size error so we can make inferences.

There are two common versions of this test, one for situations where there is a set of guidelines or percentages that your sample data should match and one where the observations themselves determine the expected counts using the independence principle. In order to make an inference about the population or populations involved, the samples used must be SRS.

Another condition that must be met is that each expected count must be at least one. Furthermore, at least 80% of the expected counts must be at least 5. Although the observed counts must be integer values, the expected counts (just like expected values) do not need to be integers. [Error alert: Many students INCORRECTLY use the observed counts instead of the expected counts to determine whether the test is appropriate.]

Depending on the type of test we are performing there will be one of two different methods for calculating the expected counts. For both types of tests, once you have found the expected counts, you calculate chi-square components for each pair of observed and expected counts:

chi-square component = (observed - expected)^2/expected. (Of course, these are all non-negative.)

You add up all of the chi-square components to get the chi-square statistic, X^2.

You compare this X^2 value to the chi-square distribution with the appropriate number of degrees of freedom to find the p-value, or probability that you could get a X^2 value at least this large, randomly, when the null hypothesis is true. If this p-value is small, we reject the null hypothesis. If the p-value is large, we do not have sufficient evidence to conclude that the alternative is preferred.

So, I haven't addressed the hypotheses. . ..

Chi-square Goodness of Fit Test (GOF)

This is the test you use to compare a sample set of observed counts to a model that is defined somewhere else by a higher authority. Some examples:

Comparing your bag of M&Ms to the distribution of colors posted on the M&M/Mars website.

Comparing your bag of M&Ms to a uniform distribution by color (1/6th of the bag / color).

Comparing the age distribution of your town to the U.S. Census proportions.

Comparing the number of students at your school making 1, 2, 3, 4, or 5 on the AP exam compared to the global distribution.

Your null hypothesis states that the distribution matches the expected distribution. The alternative is that the distributions do not match. It is important that you write the first statement in context.

If the null hypothesis says something like p1 = p2 = p3, the alternative hypothesis SHOULD NOT be "p1 is not equal to . . . " because some of the pairs of proportions could still be equal yet the numbers do not match the distribution you wanted. Instead, use verbal descriptions like the distribution does not match the model.

To find each expected count, you take the proportions from the higher authority and multiply them by the total of all observations. You will generally get non-integer values.

Check the expected counts to make sure that all of them are at least one and check a second time to make sure that at least 80% are 5 or more.

Perform the calculations described, computing the chi-square components, adding them up to get the chi-square statistic, using that statistic to find the p-value, and making a decision in the context of the problem. If you choose to reject the null hypothsis, go back through the components to find the greatest contributor to the high chi-square statistic and cite that in your decision.

Chi-square Tests of Association: Independence and Homogeneity

When you have two or more samples from one population or two or more samples from two or more populations that you are comparing against each other with respect to categorical variables you will generally perform a chi-square test of association on the two-way table that you create to summarize the samples.

Use the words of the problem to generate the Ho and Ha for this test. The null hypothesis will customarily follow the pattern there is no association between [characteristic one] and [characteristic two].

The method for finding the expected values is different from the method described for the goodness of fit test. Otherwise, the tests are virtually the same.

To find each expected value for the cells of the two-way table, multiply the row total by the column total for that cell and divide by the grand total. Again, you will likely get non-integer numbers. Check the expecteds to see if they are at least 1 and at least 5 as described above. Calculate the components and chi-square statistic using the same formulas as in the goodness of fit test and evaluate the statistic in the same way.

Setting up the hypotheses

One of the hardest problems for students seems to be figuring out what the null and alternative hypotheses should be. Consider the test itself. Whenever the observed count matches the expected count you get a chi-square component equal to zero--something that does not contribute anything to our chi-square statistic. If ALL of the numbers matched, then our statistic would be zero and it would be graphed on the far left end of our distribution, leaving 100% of the probability to the right--a p-value of 1. (Fail to reject the null!!!!)

On the other hand, if our observed values are far from the expecteds, then the chi-square components will contribute to a larger statistic and, ultimately, a smaller p-value. (If p is small enough, reject the null!!!!!)

How does this help us to generate our hypotheses? For our null hypothesis, our observeds must be close to our expecteds. When does that happen? When our idea of what should have happened actually DID happen, for instance, when we expected the distribution to be practically uniform and it was.

This is just a little trickier when we are talking about association. The null hypothesis is that the characteristics listed along the top of the two-way table have nothing to do with the characteristics listed on the side of the table. If we proposed that video-gaming and gender were independent, then we would expect the same proportion of boys to be gamers as the girl gamers. Even though the wording of the problem may be ambiguous (Are gaming and gender independent? vs Is there a relationship between gaming and gender?), the test is still the same. The comparison that you make is between the observed counts and what the counts should be if the two characteristics are independent.

Reviewing concepts

2007-03-29T15:52:00.000-04:00

The problem for Thursday's HW:
P-hat is 0.3, Ho: p = .25, Ha: p > .25

Your rival thinks that the sample indicates over 25% support for his program. He found 12/40 customers liked the idea. Write an email to the boss to enlighten him/her.

**********
The card problem:
Three cards are in a hat. One is white on both sides, one is red on both sides, and one has one white face and one red face. The cards are mixed and one is drawn from the hat and placed face down on the table without showing the underside. If the face up is red, what is the probability that the other face is also red?

Chapter 12 Inference for proportions

2007-03-12T08:14:00.000-04:00

Statistics in action. . .

Here's the basketball video.
http://viscog.beckman.uiuc.edu/grafs/demos/15.html

NCAA Brian's out in front with no way for anyone to catch up (I think). Pretty amazing. http://linnerstats.mayhem.sportsline.com/e
You'll need the password, which tells you who I think will win: gogators

Please try out this quiz and let me know how it works for you.
http://www.proprofs.com/quiz-school/quizview.php?id=567 :Basic stuff quiz

http://www.proprofs.com/quiz-school/quizview.php?id=585 :Which test do we do?

Cool sites for playing with proportions:
http://http://www.ltcconline.net/greenl/java/Statistics/HypTestProp/HypTestProp.htm

http://www.math.csusb.edu/faculty/stanton/m262/proportions/proportions.html
List of top engineering schools for recruiting as discussed in class (not in any particular order):
Cal Poly, Penn State, Penn, MIT, Florida A&M, Florida, RPI, Morgan State, Maryland, UCLA, Virginia, VA Tech, Iowa State, GA Tech, Howard, Colorado, Arizona, Cal – Berkley, North Carolina A&T, Puerto Rico, Michigan, Carnegie Mellon, Ohio State, Purdue, Illinois, Cornell, Texas, Texas A&M, Stanford, USC

This chapter is more of the same methods we saw in the last two chapters. You perform hypothesis tests and confidence intervals for proportions and for differences between proportions.

The tricky bits: (1) you have to keep track of which version of the proportion you will use for testing assumptions and for calculating standard deviations/std errors. Simply use the "best" information available. (2) Recognize when the inference is about proportions and when it is about measurements (chapter 11 methods). If you use X when you should have used p you let the reader know that you are confused.

When you have a 1 proportion hypothesis test, you have a hypothesized value for p that you use for both checking assumptions (conditions) and calculating the std dev.

When you are constructing a 1 proportion confidence interval, use the best info you have--the sample proportion. This is the lucky case where you just record the number of successes and the number of failures when you are checking the conditions. Because the estimator is used, we call the sqrt(p-hat(1- p-hat)/n) the standard error. Estimate------>>>>std error.

When you have a 2 proportion hypothesis test and you are testing to see if the two proportions are the same, well, doesn't that mean that the two proportions that you use in the std error calculation should be the same? In this case you generate a "pooled" estimator (Pooled sample proportion = sum of x / sum of n)to use for condition checking and for std error calculations. When checking conditions, use the pooled proportion * each value of n and (1 - the pooled proportion) * each value of n and make sure that each product is 5 or more.

On the other hand, when you are creating a 2 proportion confidence interval for the difference, you are not assuming that the proportions are the same, so the proportions must be checked separately and the formula for the std error resembles the formulas for two-sample conf interval std errors from Ch 11 a little bit. Checking conditions: for each sample check p-hat for that sample * sample size and (1-p-hat for that sample) * sample size.

Chapter 11 Inference for Distributions

2007-02-21T19:23:00.000-05:00

To understand this chapter you have to understand the processes of Chapter 10.

The t-distribution is a lot like the normal (z) distribution. It is much more forgiving (look for the references in the book to robustness) than the normal and we use it mostly when we have only a sample to work from--no population standard deviation.

The formulas involving t start out a lot like the z formulas.

t-statistic = (x-bar - mean)/(sample std dev/sqrt n)

and t-interval boundaries are x-bar +/- t* (sample std dev/sqrt n)

We use n-1 degrees of freedom because we "lost " one when we used x-bar to create the estimator s.

The sample std dev / sqrt n is called the standard error of the mean.

The value we use for t*, in fact the line of the table we use when considering probabilities, is based on the number of degrees of freedom (df). You can't use a line with a df = some number if you don't have at least that number of degrees of freedom. It's kind of like buying stuff. If you don't have the money, you can't buy the product. Do you realize what this means??? If you have 990 degrees of freedom and the closest choices in the text are 100 and 1000, you are supposed to select the conservative number, the one you can afford, 100 df. Now, if you can get a closer number from your calculator, use it.

How can you get the value from your calculator? (1) Use the Inv T program or function. Ti-84s with system 2.41 have it. If you have an '84, upgrade your system. If you have something else, get the program.
(2) Use the trick we demonstrated in class: Use T-INT with x-bar = 0, sx = sqrt of n, and n = n. The upper bound of the interval you generate is the estimate for t*.

Paired t-test

This is a routine t-test that is done on matched-pairs data. When you can load the first data set into L1 and the second into L2 and the following two conditions hold, you are looking at a matched-pairs design. (1) Each row of the data has to be naturally linked, as in data coming from the same person--and a different person from the rest of the rows. The two lists are DEFINITELY NOT independent of each other. (2) The variable of interest is the difference between the two values, like L1 - L2. The null hypothesis is usually mu(of the differences) = 0.

To perform the test, just do the regular t-procedures on the column of differences. DF still equals n-1.

If the two sets of data are two independent samples, that's something different. . ..

Two-sample tests

Note: The t-statistic for the difference betwen two means IS NOT t-distributed, but it is pretty close under most conditions.

We use two-sample procedures when we are looking at two separate, independent samples and trying to make an inference about the difference between the two population means.

While most of the procedure is intuitive, the standard error and the number of degrees of freedom require a little explanation.

Std Error of the difference of the means:
Do you remenber how we can't add std deviations? And how the variance of the difference of two variables is the sum of the variances? Put it together for this problem.

Find each sample variance--(s/sqrt(n))^2. Add the two sample variances together. Take the square root. In these formulas, s1 is the sample std dev for the first sample, n1 is the size of the firs sample, etc.

Then the std error of the difference = sqrt( (s1^2/n1) + (s2^2/n2) ).

Degrees of freedom:
For the number of degrees of freedom, either use the number that the calculator or the computer calculates for you or use the more conservative minimum of n1-1 or n2-1.

Hypothesis:
Ho: mu1 = mu 2 which is equivalent to Ho: mu1 - mu2 = 0

Other than these little changes, the procedures are similar to those you've already practiced.

Pooled vs unpooled

This refers to the situations when you believe that the variances of the two populations should really be equal. Using a concept similar to our Law of Large Numbers, combining the standard deviations from the samples in a clever way creates an even stronger estimate for the ONE estimated standard deviation. This is pooling of variances.

Just because the means are the same we cannot assume that the variances are equal also.

We almost never pool variances of X-bar. You can generally leave your calculator set on UNPOOLED and forget about memorizing the formula. You can only pool variances if you are really sure that the variances are equal.

Chapter 10 Beginning of Inference

2007-02-06T20:43:00.000-05:00

This chapter introducecs important methods under the highly unrealistic conditions where we know the population standard deviation but not the population mean.

Point estimates for the average value of X found through samples are generally good estimates, but they are wrong. You can generate a better estimate by creating a confidence interval.

The confidence interval =
x-bar +/- Z* times sigma of x / (Sqrt n).

We get Z* from the t-table for a specific confidence level, for instance when we want a 95% we use 1.96.

In creating a complete solution we first write down all of the given information. Define your variable. Then we determine whether the central limit theorem has "kicked in" or if the underlying data were already normally distributed. Be sure to address whether the data were from a SRS. Graph them if you have them to make sure there are no gaps or outliers. Is the sample size less than 1/10 of the population size???

Identify what you are trying to produce-- a 95% Z interval for mu and give the formula. Show how the numbers are plugged in and calculate the interval.

Write the interpretation of your interval.

We are 95% confident that the true population mean value of [insert the contextual information here] falls between [lower bound] and [upper bound].

Refer to your notes for all of the baaaaaaaaaaaaad interpretations of a confidence interval and NEVER use them. :)

If a value of mu had been proposed before we collected our sample, we could see if the value falls within our interval. If the proposed value does fall in the interval, then it is a reasonable value, although not necessarily correct. If it does not fall in the interval, then it is not a reasonable value according to our sample values.

Hypothesis tests

For hypothesis tests, you develop a null and alternate hypothesis BEFORE you collect data. Both hypotheses use the parameter (NEVER THE STATISTIC) and they are considered logical opposites. The null hypotheses ALWAYS has an "equals" aspect to it: the alternate hypothesis is always <, >, or not equal to.

For instance: H0: mu = 15
Ha: mu > 15.

Although these are not actually opposites, finding evidence that mu is less than 15 provides no support of the alternative hypothesis. You can think of the null hypothesis in this case as mu<=15, which still has an "equals" in it. This is the way I learned hypothesis-writing back in the day and it is still acceptable, but not as common.

Alpha, Beta, Type I error, Type II error, and Power

Alpha is the likelihood of a Type I error--accidentally rejecting the null hypothesis when it was actually correct. (Like convicting the wrong guy.)

Beta is the likelihood of a Type II error--failing to reject the null when it was wrong. (Kind of an error of omission, or not enough evidence to convict.)

Power is the likelihood that the test would have been sensitive enough to pick up the difference between the hypothesized mu and the actual mu (given some other new value for mu). This is the complement of Beta. Yes, 1 - Beta = Power. 1 - power = beta. Power + beta = 1.

Notice that alpha and beta are NEVER added together. They don't live under the same conditions--one assumes that the null was true and the other that the null was false. DO not fall into the trap of EVER adding alpha and beta together (unless you are TOLD to do it and then only if they offer you a lot of money or a passing grade on a test).

Calculating beta is easier than people think on the calculator.
(1) Figure out what the critical values are for rejection of Ho in terms of x-bar.

(2) Find the area under the curve centered at the NEW mu that falls between these critical values. You can use normalcdf(left_critical_value, right_critical_value, new mean, standard dev or error of x-bar).

Chapter 9 - Sampling Distributions

2007-01-20T18:43:00.000-05:00

How does the sample size affect our estimate and our decisions?

Parameters are the (usually unknown) measures of a population. Often they are represented by Greek letters like mu and sigma.

Statistics are the calculated measures generated from the samples. Statistics are estimators for parameters.

When the average of a statistic is the parameter itself, it is called an unbiased estimator. X-bar is an unbiased estimator for mu, the population mean.

Sampling distributions are the distributions of all of the averages of all of the samples of size n taken from a population.

When the sample size n increases, the variability of the means of the samples decreases--the graph of the sampling distribution is taller and narrower. When the sample size n decreases, the variability of the means of the samples increases.

This holds for sample proportions. The mean of the sample proportions (p-hats) is the true proportion for the population, p. Under special conditions we can use a formula for the standard deviation of the p-hats: SQRT(p*(1-p)/n).

The condition that allows this is that the sample is less than 1/10th of the population (and, of course, we're talking about simple random samples!!)

Also, the really BIG twist is that we can also use an approximation to the normal distribution when the expected numbers of successes and and failures are both 10 or more.

So, about that CLT thing. . . What was the REALLY BIG idea with the Central Limit Theorem???

How do you express the distributions for a binomial X, a geometric X, a uniform X, a normal X, the sampling distribution (X-bar), and the sample proportions (p-hat)?

When can you assume that the sampling distribution is approximately normally distributed?

What do you have to write to support your calculations of mean and standard deviation? your calculations of probabilities?

Chapter 8 - Binomial and Geometric distributions

2007-01-08T17:02:00.000-05:00

Part 1 - Binomials

Binomial distributions have the following defining characterisitics:

(1)Only two mutually-exclusive and complementary events are possible on each trial--success or failure.

(2)The number of trials is fixed (n).

(3)The probability of a success on any trial is fixed at p. This DOES NOT mean that the probability of a success is always 50%.

(4)The trials are independent--knowing one outcome does not help you predict the next.

Always define what X represents, for instance, X = number of daughters (successes).

Shorthand identification for a binomial distribution: Binom(n, p).

The calculator will provide probabilities given n and p: binompdf(n,p[,x]) and binomcdf(n,p[,x]). Use pdf when you want probabilities for individual values of X and cdf when you want cumulative values, like the probability that the number of successes is less than or equal to 5. Insert the X value when you want just one value for a specific value of X. You may omit it when you want all the probabilities. Caution! For binomials, the least value X can take is ZERO, not one, so make sure that you associate the right X values with te correct probabilities.

The formula for P(X=k) = nCk p^k * (1-p)^(n-k).

nCk is "n choose k" or n!/(k!*(n-k)!).

If you calculate these probabilities for each possible value of x from 0 to n and add them up you will get a sum of 1.

The expected value or mean of the number of successes in a binomial setting is "mu sub x" = n*p.

The variance of the number of successes in the binomial setting is sigma squared sub x = n * p * (1-p).

The square root is (of course!) the square root of the variance.

What were those directions for loading binomial values into the lists and graphing as histograms? Use seq(X,X,0,n) --> L1 to populate the Xs and binompdf(n,p) --> L2 to insert the corresponding probabilities. To graph, select the histogram tool, use L1 as the xlist and L2 as the freq. You can use zoom 9 to generate a first stab at the graph. Then fix the graph using the window controls.

Part 2--Geometric Distribution

This was different from the binomial in that we are counting the number of trials UNTIL we achieve success, then we stop. This means that X is the number of trials it took and there is no "n" involved. Theoretically, it could take us infinitely many tries before we had a successful result.

Defining characteristics: fixed p, s/f, independent trials, count until success (not a fixed n).

The expected value of x, the number of trials required, is 1/p, where p is the probability of a success in one try. The variance is (1-p)/p^2.

The probability distribution for x = 1, 2, 3, 4, etc. is p, (1-p)p, (1-p)^2*p, (1-p)^3*p, etc.

What is the probability that it takes more than k attempts before you get a success?

State of Fear

2007-01-02T10:42:00.000-05:00

The rhetorical questions:

What does State of Fear refer to?

What is true? How do you know? Who do you trust? What role does the statistician play in your understanding of news? What role do the media play?

The question to answer:

How can someone lie with statistics?

Reviewing for the 1st Semester Final

2006-12-15T16:53:00.000-05:00

Heard in passing today:

Hang up. Log off. Study.

OK, if you log off you can't blog with the APSTAT crowd.

*** What are the most important concepts from each chapter?
*** What are the parts that you still don't understand?

HW: Write down the three most important from each chapter, 1-7. Due Monday.

*** CiCi's on Sunday.
*** Review sessions Monday 8-9PM for 5th and 6th periods, Tuesday 8-9 PM for 1st and 7th periods.

*** I have a meeting Monday morning. I will get to the MU as soon as possible.

*** Yes, we are having classes in 1st and 7th periods on Tuesday!

Post away, my statty friends.

Chapter 7 Random Variables

2006-12-07T18:53:00.000-05:00

This chapter prepares us to work with distributions of random variables and to find their measures of center and spread.

E[X] = the sum of (x * P(x) for all values of x).

The rules for means are straight-forward. The expected value of a random variable, E[X], is the mean, commonly called mu. The mean of the sum of random variables is the sum of the means. The mean of the difference of random variables is the difference of the means. The E[aX] = a*E[X]. The expected value of a constant is just that constant.

Really complex example: E[aX + bY + c] = a*E[X] + b*E[Y] + c.

When you work with measures of spread you have to be more careful! You cannot add standard deviations. You must work with their squares--the variances.

Var[X] = the sum of ((x-mu)^2 * P(x) for each value of x)

= E[X^2] - (E[X])^2

The Var[aX + b] = a^2 * Var[X]. The constant, b, does not vary, so it contributes NOTHING to the variance.

Now, IF X AND Y ARE INDEPENDENT (THAT"S A BIG IF!!!!!!), then Var(X + Y) = Var(X) + Var(Y). If they are NOT independent, then there is some covariance factor which could be increasing or decreasing the variance. The covariance concept is beyond the scope of this course.

That covariance thing is why we can't calculate the variance of the sum of the math and verbal portions of the SAT directly. We know that these scores are not independent.

Examples from class:

X={1, 11}, Y={-4, 20}, X+Y={-3, 7, 21, 31}

Find the variance of each set and look for a pattern.

Here's another:
X={1, 15}, Y={-4, 44}, X+Y={-3, 11, 45, 59}

Can you create two sets which, when added together, have a variance of 100?

Chapter 6 Probability

2006-11-13T15:39:00.000-05:00

GO TO THE CLASSHOMEWORK SITE TO PICK UP DAILY HOMEWORK ASSIGNMENTS>>>>>>>>>>>>>>>>>>

There's a Homework link on the menu to the right that will take you there!>>>>>>>>>

Essential questions:
Is probability a fixed number or something developed through many, many repetitions?

How can a probability model help us to make decisions?

6.1 Randomness

The chapter starts off with some philosophical and theoretical concepts that you probably didn't consider when you first took probability and developed that deep, underlying appreciation for all things probabilistic.

What is probability? There are two positions on this subject. First, you have the experts who believe that there is an intrinsic probability associated with a random phenomenon. For instance, the actual probability of flipping that quarter in your pocket and getting heads is some fixed number between 0 and 1. All the observations we get from flipping that coin ba-zillions of times will only point us in the direction of the true probability of a success.

There was an important piece of information in that last bit: a probability p must fall between 0 and 1 inclusive. [0<= p <=1]

Then you have the other camp: the experts who claim that all the possible flips of the coin define the probability of success for that coin. Of course we can never observe ALL possible flips of a coin, because every second it is not being flipped is a waste of a flip! This is consistent with the authors' approach to this chapter of the book.

OK, I guess that there would be exceptions. You wait eagerly while the conveyor belt at the U S Mint carries a bright shiny quarter to you. It falls off the production line into your hands, you flip it in the air where it glistens and falls, heads up, to the floor of the Mint. A bulldozer appears out of nowhere and smashes the quarter into a mangled silver mess of metal. The probability of getting heads on that coin WAS 100%. That information doesn't do us much good now.

Back to the chapter.

First things first. Just because there are two possible outcomes to a random phenomenon does not mean that you have a 50% chance of a success.

The authors define probability as the proportion of times the outcome would occur in a very long string of repetitions.

Independence means that one trial is not going to influence the outcome of any other trial. If the outcomes are determined by some non-random influence, then it is not a random phenomenon.

But you know that you've seen problems that dealt with events that are not truly random--like whether or not a student took the SAT prep class. The way that this non-random event is turned into a random event is by asking what the likelihood of randomly drawing a student who HAD taken the SAT prep class was. If it's not random, then we don't have a probability distribution.

So, what good is running a computer simulation? You just have to give it the answer-- the probability of a success--and in the long run it would tell you that you had that percent of successes! That's if you're lucky. In the shorter run the computer can help us measure how likely or unlikely a particular outcome from a random even would be, given that the true probability of a success was some number p.

6.2 Probability models

Sample space is the list of all possible outcomes. For instance {heads, tails} or {H, T} is the set of all possible outcomes from one flip of our trusty quarter. For two flips the outcomes could be order-based (HH, HT, TH, TT} or summarized {2H, 1H1T, 2T}, depending on what you are trying to count. Note that all the outcomes in the summarized set are NOT equally likely. Likewise, the outcomes from adding together the number of pips from the roll of two dice would give the sample space {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} with probabilities 1/36, 2/36, 3/36, 4/36, 5/36, 6/36, 5/36, 4/36, 3/36, 2/36, 1/36 respectively.

When we relate the probabilities to discrete outcomes we create a probability distribution. This is often represented in a table. If the data are continuous, there will be a function that describes the probability density.

Some basic concepts from counting theory and probability:

tree diagram- you can map out what happens at each stage of a multi-step random process. Multiplying probabilities as you move out toward the branch of the tree will yield the joint probabilities. The sum of all of the joint probabilities will be 1.

multiplication principle- to find the number of ways joint, independent events can be combined, find the product of the number of ways each step can be performed. For instance, 3 shirts with 5 pants means 15 different combinations if we don't care what goes with what.

replacement- if you put the selected object back into the pool of objects to draw from. If you pick a card, record it, put it back, and draw again, the make-up of the deck does not change from draw to draw. That is WITH replacement. If you kept the card out of the deck, then the remaining cards do not have the same distribution as the original deck. That is WITHOUT replacement. RandInt is with replacement: sorting L2 and L1 by rand numbers in L2 is without replacement.

event- an outcome or a set of outcomes of the random procedure or phenomenon.

We use the expression P(A) to mean the probability that A occurs.

0 <= P(A) <= 1, like we agreed before.

Since the sample space S contains all of the mutually-exclusive, exhaustive possible outcomes of the phenomenon, P(S) = 1.

The probability that an event A does not happen is 1-P(A). There are many symbols for this event, the complement of A. Since they do not translate well to the html format, write them down in class so you will recognize them.

If two events A and B cannot both happen together, they are called disjoint and the probability of at least one of the events happening = P(A or B) = P(A) + P(B). If they actually had some overlap, like drawing a queen and drawing a spade, then you would have to subtract out the probability of the overlap (the Queen of spades!). This more general formula is P(A or B) = P(A) + P(B) - P(both A and B).

You can use a Venn diagram to represent these relationships and make the procedures clearer.

If all outcomes are equally likely, then the probability of any one happening is 1/(the number of outcomes). This is like our treatment of problem 6 in the book and the activity with the numbers 1-2-3 in class. Each outcome from problem 6, {H H H H}, {H H H T}, {H H T H}, {H T H H}, {T H H H}, {H H T T}, etc., is equally likely. We know that there are 2 X 2 X 2 X 2 = 16 possible outcomes (by the multiplication principle). Then the likelihood of any specific event happening is 1/16. If we know that there are 4 ways to get exactly one tail, we can combine these probabilities to get the probability that any one of those four outcomes happens, P(exactly one tail), = P( {H H H T} or {H H T H} or {H T H H}or {T H H H})=4/16.

It is usually wise to write all the probabilities with a common denominator so you can check tht the sum is 1.

Independent events revisited: In Chapter 4 we looked at the probability rules surrounding independent events. If two events are independent, then one event happening does not affect the probability that the other happens.

For instance these are NOT independent:

A = being a Lassiter student
and
B = owning a piece of Lassiter spiritwear

If you are a Lassiter student, then you are FAR more likely than other people to own Lassiter apparel. If you do not own Lassiter apparel, then you have a much higher likelihood of going to Kell.

C= being a Lassiter student
and
D= not being a Lassiter student

If one of these is true, then the other CANNOT be true. Therefore, the occurrance of one SEVERELY impacts the likelihood of the other. These are mutually-exclusive. Mutually-exclusive events are NEVER independent of each other.

It takes something special to be independent. The two most common ways to prove independence are

(1) to check that the product of the marginal probabilities equals the joint probability. If P(A) * P(B) = P(both A and B), then events A and B are independent. This is really an if-and-only-if statement.

and

(2) to check that the marginal probability equals the conditional probability. If P(A) = P(A|B), then B's happening does not affect A's likelihood and A and B are independent.

Remember: P(A) * P(B) = P(A&B) WHEN A and B are independent.

Also, coins, dice, cards, and the Roulette wheel have no memory. They don't care what their last outcome was: every trial is independent.

6.3 More

The 5 basic rules of probability are recorded on page 341.

Can you answer. . .

what is a union of events?

how do you compute the probability of at least one of some collection of events happening?

how is the addition rule modified when there is overlap between the events?

how is the addition rule modified when there is overlap among three events?

what does conditional probability mean? Can you interpret a Venn diagram to calculate a conditional probability? Can you CONSTRUCT one??????

what is the intersection of two events?

when is the probability of BOTH of two events equal to the product of their respective probabilities? when is it NOT?

can you read tree values with probabilities to calculate marginal, joint, and conditional probabilities? Can you CONSTRUCT one?????

It looks like we need better understanding of INDEPENDENCE.

Problem 6.36 If the probability that the woman in the study was over 65 years old was (.365 + .190) = .555 and the probability that she had the tests done was (.321 + .365) = .686, then the probability that she was over 65 and had the tests done should be .555*.686 if the AGE and TEST status are independent. We multiply these two marginal probabilities together and get .38073. We look in the table and see that the actual probability of being over 65 and having the tests done is .365. Because these are NOT the same, we conclude that there was some connection between the two characteristics, AGE and TEST status.

Problem 6.42 d. You have to demonstrate that the two characteristics are not mathematically independent. Find P(widowed), P(65+), P(widowed)*P(65+), and P(widowed AND 65+). If P(widowed)*P(65+) = P(widowed AND 65+), then the two characteristics are independent. You HAVE TO show the numbers. YOu have to show that they are equal--or not.

To find each piece of the puzzle:
P(widowed) = row total # widowed/total of all women

P(65+) = column total # of 65+/total of all women

P(widowed and 65+) = number from the body of the table where widowed and 65+ intersect/total of all women.

6.44 P(W) = 856/1626, P(W|Pr) = 30/74. Because these are not equal, we know that the characteristics Female and Professional are not independent. We prove it by comparing P(w)*P(Pr) to P(W and Pr). (856/1626)(74/1626) does not equal 30/1626. Therefore, they are not independent.

6.46 P(Male) = 24,457/(24,457+6027), P(F) = (15802+2367)/(24457+6027), P(F|Male) = 15802/24457, P(F|Female) = 2367/6027

"Among those who. . .males are more likely than females . . ."

6.47 P(all three)=5%, P(Coffee only) = 20%, P(coffee and tea only) = 10%, P(tea only) = 5%, P(Tea and cola only) = 5%, P(cola only) = 15%, P(Coffee and cola only) = 20%. So, what is the probability that a randomly-selected adult drinks none of the above?

6.48 P(B|A) = .32 = P(A&B)/P(A) = P(A&B)/.46

Solve for P(A&B).

6.49 P(R|F) = .8 = P(R&F)/P(F) = P(R&F)/.4

Solve for P(R&F)

Interesting site: http://www.paly.net/~sfriedla/apstatistics/

Chapter 5 Sampling, experiments, and simulation

2006-11-01T22:41:00.000-05:00

Essential questions:

Can the data we collected be generalized to the population?

How can the survey or experiment be designed to accomplish our goals?

How can we confirm our suspicions using simulation?

-----------------------------

Running list of key concepts from class:
Survey
Census
Simple Random Sample (SRS)
Systematic Random Sample
Stratified Random Sample takes samples from all strata
Convenience Sampling
Table of Random Digits

Cluster Sampling takes a sample from a few clusters
Multi-stage sampling is a complex form of cluster sampling
Probability Sample like the computer lottery at LHS
Bias when method favors certain outcome(s)
Undercoverage when systematically omits part of population from inclusion
Non-Response when they refuse to participate
Sampling Frame is the list from which the sample is drawn

Experiments:
observational studies
experiments
experimental units/subjects
treatment
factor
level

control
comparison of several treatments
placebo effect results in bias
reduces the effect of lurking variables (confounding and bias)
could include blocking (not required) *BLOCKING reduces the variability within the group, so effects of the treatments can be more easily recognized.
control group
matched pairs design is smallest block

randomization
matching of characteristics does not work
required real randomization, not just haphazard guesswork
makes the effect of any uncontrollable lurking variables affect all groups equally, thereby also reducing bias
When the problem asks for the experimental design, it requires that you describe how you will randomly allocate experimental units/subjects to treatment groups. Two key points to remember: you CAN'T randomly assign subjects to blocks, because the characteristic you are blocking for is not random, AND this is not a SRS.

replication
allows you to generalize your data to your population
makes the experiment more sensitive to differences among treatments, instead of just random variation between the groups. The compiled or averaged results from a larger group of subjects should more precisely represent the actual, underlying truths of the relationship than results from smaller numbers of subjects. Of course, there is a cost trade-off.

simulation
use table of random variables or random number generator
CLEARLY identify what specific random outcomes represent, such as
The digits 0-4 represent a vote for Adams, 5 & 6 are a vote for Jefferson, 7-9 will be a vote for Roosevelt. Take one random digit at a time, comparing the result to our mapping above, until we have identified 100 votes and the corresponding candidates.

or . . . in cases where you CAN'T reuse a number . . . "Assign each child a unique number 01-47. Take two digits at a time from the TORD (table of random digits), recording the names of the students as we select their number, throwing out any number greater than 47 or those which have already been used.

When a question asks you to describe or explain, there should be a description or explanation in your answer. Just providing a mapping is not sufficient.

When it asks you for the sampling or experimental DESIGN, an explanation of how you are going to select your random units must follow. You must describe how you will assign the digits to the outcomes, how you will take the digits from the TORD, what "toss out" rules you need for duplicates or numbers that have no correspondences, and when you will stop. You have to explain it all. You will need to write.

Some common calculator stuff: Rand(100) selects 100 random numbers between 0 and 1 where repetition is HIGHLY unlikely.

RandInt(5,29,31) selects 31 random digits from the range [5,29] and allows repeats.

SortA(L2,L1) sorts both L2 and L1 in the ascending order of L2.

Watch this space for more key words.

Chapter 4 Nonlinear relationships

2006-10-07T14:09:00.000-04:00

Read through the list of goals at the end of the chapter frequently.

In this chapter you will work with bivariate quantitative data and relationships between two categorical variables. For the quantitative part, you will learn to "straighten" x-y data, that is to use a transformation function to create a new relationship between f(x) and g(y) that is approximately linear. Find the least-squares relationship between the transformed data, then find the inverse of the original transformation function to transform the model into a curve which passes through your original x-y data. It's pretty cool to accomplish this and magnificently powerful math.

The second part, the categorical part, covers conditional and marginal probabilities. For instance, break the class into m/f and soph/jun/sen identifiers. Each person falls into exactly one of the gender groups and exactly one of the class year groups. Overall, what is the likelihood that a randomly-selected person is in a particular class? What is the probability that they are a particular gender? If they are a girl, then what is the probability that they are a senior? If they are a senior, what is the probability that they are a guy? Also, if guys do better than girls in 1st period and guys do better than girls in 5th period, how could the combination of the two classes indicate that girls are doing better than boys?

Dress appropriately for the weather and for doing activities that involve sitting on the floor this week. See you 10/8 at CiCi's???

Chapter 3 Linear relationships

2006-09-30T21:05:00.000-04:00

Now you've done it all! Can you

identify bivariate data?

graph response and explanatory variables?

differentiate in a scatterplot for a categorical variable?

describe data represented in a scatterplot?

find the least-squares regression line?

compare and contrast the concepts of regression, correlation, association?

explain what the correlation coefficient tells us?

explain what the coefficient of determination tells us?

use a predictor line (LSRL) to predict the value of y for a given x?

use a predictor line to calculate and interpret residuals?

calculate EVERYTHING using the formulas in the text?

explain the vocabulary?

identify the key topics from this chapter?

write good questions for a test?

teach someone else how to work these problems?

squeeze the maximum information from real data using linear methods appropriately?

prove (in writing) that you understand and can apply the concepts of this chapter?

What have I left out?

CU @ CiCi's.

Chapter 2 thoughts

2006-09-20T18:04:00.000-04:00

Essential question: Why is the Normal distribution so special?

Here's a teaser for you--if you know that 20% of the data in a normally-distributed population fall below the x-value 306 and 80% fall below the value 772, can you find the mean and standard deviation of the population???

I will be in the classroom as early as possible on Thursday, but I have a parent meeting at 7:45. Please use your other resources well. Don't forget the book! The chapter summaries are great tools. Do the practice quiz online. Dream up questions that I might ask.

Have a slice of pizza for me.

Pre-test discussion

2006-09-11T18:04:00.000-04:00

Ask and answer the most pressing questions here. You WILL nned to establish an identity to post. Please DO NOT use your first and last names.

Have you taken the practice quiz yet? The link is on the right edge of the main screen. Spread the word to your friends and neighbors.

Good luck.

Straightening data

2006-08-31T22:09:00.000-04:00

Here's the plan we developed in class:

Load your (x,y) data into L1 and L2.
Look at them.

If they are not already straight, figure out if the ideal model would pass through the x-axis, the y-axis, or both. To save some time, try this to straighten your data:

Case 1: if the ideal model would cross the y-axis, take the ln of the observed y values. Case 2: if the model would cross the x-axis, take the ln of the x values, too.

THEN
Case 1: Run the linear regression on the original x and the ln y. Change your stat plot to show the (x, ln y) points with the linear regression equation. If this is a good fit, then the residuals will be scattered. Correct the linear regression equation to reflect that the y-values were really ln y. Solve the fixed equation for y.

Case 2: Run the linear regression on the ln x and the ln y. Change your stat plot to show the (ln x, ln y) points with the linear regression equation. If this is a good fit, then the residuals will be scattered. Correct the linear regression equation to reflect that the y-values were really ln y and the x-values were really ln x. Solve the corrected equation for y.

To check your results, put the new equation for y into the y= register to graph. Change your statplot to show the original data (x,y), probably in L1 and L2. The curve you generated should pass neatly through the data.

Note that this method finds the line which miniomizes the sum of the squared residuals from the STRAIGHTENED data, not the squares of the residuals from the curved fit.

Oh yeah, go Braves!

Welcome to the new school year

2006-08-15T19:19:00.000-04:00

How is a permutation different from a combination? How are they similar?

Scores from the exam and a Must-see-video

2006-07-02T21:04:00.000-04:00

I'd love to hear how you did on all of your AP exams and how your summer is going, but PLEASE do not use first and last names on the BLOG. YOu can send private emails to my school email if you want.

You guys are great!

I hope that you are done with your summer assignments so you can enjoy the last month! :)

You HAVE TO check out this video:
http://video.google.com/videoplay?docid=5243677894327730537

Let me know how you like it.

Mrs. L

AP Central link

2006-04-28T16:47:00.001-04:00

The website you will go to is
http://apcentral.collegeboard.com/exam/0,3060,152-0-0-0,00.html

If you have never been there before, you should register as a student so you can look at info about all of your tests. Also, there is a section on tips for students by Darren Starnes in the Stat part that you should read.

C U @ C CCCCCCCCCC.

Chapter 13 Chi-square tests

2006-03-23T22:44:00.000-05:00

There are three different tests in this chapter, but only two distinct methods.

The first method is what you used in class to determine whether your sample was reasonably consistent with the hypothesized proportions by color of Goldfish, Froot Loops, or Smarties. You determine the expected counts by multiplying the hypothesized proportion by the total of objects. The number of degrees of freedom is the number of categories minus one. This was a Chi-square goodness of fit test.

The next method you will use is the Chi-square test of homogeneity. This is used when you have two populations that you are comparing to see if they have a common distribution by the categorical variable. You base this decision on your sample comparison. Using the same methods, you can perform a Chi-square test of independence. This is used to determine whether a sample described in a two-way table by two different characteristics demonstrates independence between the two variables or if there appears to be a connection. In these cases, you have to multiply the row total by the column total and divide by the table total to get the expected count for each cell. You will use (r-1)*(c-1) for the number of degrees of freedom. This is the number of cells you would have to fill in (if you knew all of the totals) before the rest of the cells' values are determined.

You have now seen every topic on the Barron's guide and on the AP exam syllabus. We're almost there!

Chapter 12 Inferences about proportions

2006-03-09T16:13:00.001-05:00

BINOMIAL CONNECTION:
The methods of this chapter are based on the binomial distribution.

Let x be the number of successes in n trials.
If the conditions of a binomial setting hold,
then mu of x = np and
sigma sub x =sqrt(n*p*(1-p)).

Now, because p-hat, the estimator for the population proportion equals x/n,
mu sub p-hat = (mu sub x)/n Which means that p-hat is an unbiased estimator of p

and

sigma sub p-hat = (sigma sub x) / n.

Well, if you take that last part, (sigma sub x) / n, and substitute for sigma sub x,
you get sigma sub p-hat = sqrt(np(1-p)) / n

which can be rewritten as sqrt(p(1-p)/n).

WHICH P DO I USE?

If you have a hypothesized p, you use that. For instance, if your previous study or some expert indicated that p = .35, then you use .35 in your hypothesis, the standard deviation for your hypothesis test, and calculations to find the minimum sample size for a margin of error.

You also use this value when checking assumptions np>10 and n(1-p)>10.

If you have only your sample proportion, then you use p-hat to estimate the standard deviation for confidence intervals and for checking conditions for CI: n*p-hat> 10 and n*(1 - p-hat) > 10.

If you have neither, then you must be finding the minimum sample size, so use the most conservative estimate: .5.

2 proportion methods:

It helps A LOT to make a table of values as they showed in the book.

For confidence intervals, the methods just as you imagined. You are developing a confidence interval for the difference between two proportions,
so use p-hat1 - p-hat2.

For the standard deviation,look to the variances. Add the variances of the two samples and take the sqrt. Among the conditions, compare the products n1*p-hat1, n1*(1 - p-hat1), n2*p-hat2, and n2*(1 - p-hat2) to 5. Each product must exceed 5.

For hypothesis tests, there is a nifty twist. Your null hypothesis probably stated that the two proportions were the same. Therefore, their standard deviations should be combined. Take (x1+x2)/(n1+n2) to calculate a new, stronger p-hat which you use for standard deviation calculations and checking conditions.

The standard deviation would be sqrt( p-hat(1-p-hat)/n1 + p-hat(1-p-hat)/n2), but that requires that you enter p-hat too many times. Rewritten, that formula is sqrt( p-hat * (1-p-hat) * (1/n1+1/n2)). It looks nicer in the book. Go there to read all about it.

There are super examples in this chapter.

Chapter 11

2006-03-03T08:10:00.000-05:00

The test is Tuesday, March 7th. I will not be available before school to help. Prepare early!

Sunday. 2-4. You know where.

Chapter 10 test results

2006-02-25T18:13:00.000-05:00

I will return tests to interested students at CiCi's on Sunday or in class on Monday. No, I do not pay students to attend CiCi's.

SCAD for statistical inference problems

2006-02-05T12:30:00.000-05:00

Georgia teens are probably familiar with SCAD-the Savannah College of Art and Design. Remembering this acronym can help you to include all of the parts of an inference problem (and maximize your points!!!!).

First you will address the SET-UP. This is where you write down all of the information that you pulled out of the question. You will define the hypotheses or the type of confidence interval, the statistics, and any parameters disclosed. You need to define your random variable. Keep in mind that mu and p do not vary; they are fixed. The values of x-bar or p-hat that you get from samples will vary. Therefore, you define your random variable x-bar or p-hat IN WORDS and symbols. Continue by identifying x-bar as the average value of ____________ from samples of size ______ or p-hat as the sample proportion of ___________________ with samples of size ______ (inserting the words you used when you defined your random variable and sample size). Define mu and p as the measures for the entire population. You can't use a symbol until you define it.

In hypothesis testing, be sure to use the values of mu or p in your hypotheses, not the statistics. For instance, if your were testing the proportion of students who lurk on the blog instead of writing to it when the experts think the proportion is 80%, but you think it is higher, your hypotheses are Ho: p = .8 and Ha: p > .8. Also, you only have an equals statement in the null hypothesis Ho.

The second portion of the complete answer is the ASSUMPTION or condition check. Yes, I know this is out of order, but at least you'll remember to do it! Most kids lose points by forgetting to do this or doing it poorly. The resource page of assumptions and tests in the appendix of the Barron's study guide provides a great summary of the conditions you need to check. Pay particular care to things like p or p-hat in the formulas. You have to use the right one to get satisfactory results. Don't just copy the items from the list and put check marks next to them. The readers know that you haven't actually done the check. Identify the reason why you did each test, like testing for 10n < population allows you to use simplified forms of the standard deviation. Know this. Write it down as your result after you plug in the values and test the conditions.

Of course the most satisfying part is the CALCULATION. Tell the reader what calculation you're doing, write out the formula, plug in the values, show how it is calculated, and write the numerical answer. Draw the picture. You can use your calculator to provide probabilities or z-values or t-values the same way you would use the standard normal or student's t distribution tables. Don't use them to magically provide the answer. You won't get any credit.

The final part is the DECISION. This would be the most important part to your employer. State your decision or explanation of the confidence interval in the terms of the problem, connnecting your numerical answers, the probabilities involved, and the actual words the author used. This is not time for fancy paraphrasing or concerns about plagiarism. The authors want an answer to their problem--not the answer to some related and colorfully-worded problem. Give them the facts, the probabilities, and a clear decision.

A local professor and excellent AP Statistics tutor, Michael Roty, once told me that his memory hook for answering statistics problems is "What did you do? Why did you do it? What does it mean?" I think that this summarizes the expectaions of the authors nicely. The SCAD structure should answer these questions.

Chapter 10 The beginning of inference

2006-01-29T13:08:00.000-05:00

What is the goal if this chapter? How can these methods be used?

On an unrelated note, visit http://www.infoplease.com/p/brainpop/basicprobability.html for a quick review of basic probability.

Chapter 9 Sampling distributions

2006-01-11T21:31:00.000-05:00

What big ideas have you identified in this chapter?

CiCi's Sunday January 23rd. 2-4

First Semester Final Exam

2005-12-19T18:06:00.000-05:00

We couldn't possibly cover all of the first semester's content in two days of review, so let's point out a few topics that no one has asked about:

Correlation coefficient r and coefficient of determination r-squared. What special insight does the value of r-squared give you about the relationship between x and y?

Why can't we just take the square root of r-squared to get the value of r?

Slope of a LSRL = the estimated increase (or decrease) in the response variable for every unit increase in the explanatory variable.

The easiest way to get it is r*sy/sx where sy is the sample standard deviation of y and sx is the sample standard deviation of x.

While we're talking about sample standard deviations. . . the formula is SQRT(variance of the variable), so the sample standard deviation of x would be

SQRT[(sum of the squares of (Xi - Xbar) for all values of X)/(n-1)]. N is the sample size.

Don't panic if you can't read that--just look up the formula in the text.

What does it mean to be resistant to outliers? Give examples of measures which are resistant. Give examples of some which are not.

What are the benefits of different types of graphs (box and whisker, stem and leaf, histogram)?

How do you know if a set of data is approximately normally distributed? Look it up.

Why do we block?

Why do we experiment?

What makes an experiment special?

What are the characteristics of a well-designed experiment?

Why do people sometimes need double-blind experiments?

What is the placebo effect?

How do you know if two characteristics are independent?

Things to think about when you should be studying

2005-12-15T19:46:00.000-05:00

The icon used for the command SAVE in Microsoft's Office applications is a 3.5" diskette. Now that diskettes are nearly obsolete, when will they change the icon and what will they change it to?

Chapter 8 Binomial and Geometric Probabilities

2005-12-13T18:03:00.000-05:00

What are the differences between **having two kids and counting x=the number of girls** and **having kids until you get a girl**? What is the random variable x in the second case? What are the means [expected values] of the random variable x for each of these scenarios? What is the standard deviation of x in the first case? How could you simulate each of these scenarios?

How are these distributions similar? How are they different?

Chapter 7 - Random Variables

2005-11-29T20:13:00.000-05:00

How do you distinguish between a discrete random variable and a continuous random variable?

Compare and contrast probability histograms and density curves.

If X is discretely distributed for the integers {1, 2, 3} and P(X=1) does not equal P(X=3), does the expected value of X have to be an integer? Why or why not? Does the mode have to be an integer? Why or why not? Does the expected value of a distribution have to be a value of x from your distribution (for instance, does the average number of pips on one die rolled have to be 1, 2, 3, 4, 5, or 6)? Does the mode have to be an observed value of x? Why or why not?

How does the Law of Large Numbers relate to the Kid-sino lab on November 18th?

The mean of the sum is the sum of the means.
The variance of the sum is the sum of the variances (if the variables are independent).
The variance of the difference is the SUM of the variances (if the variables are independent).
Why?

The variance of 2X is 4 times the variance of X.
The variance of (X + Y) is the variance of X plus the variance of Y (if the variables are independent).

Why are these different formulas? Or are they?

Have a super day.

State of Fear

2005-11-17T15:19:00.000-05:00

Speak your mind, but don't spoil it for the rest of the readers.

Chapter 6 - Probability

2005-11-07T14:56:00.000-05:00

Alas, here's your chance to finally learn to like probability. We'll be covering the important stuff and giving you the opportunity to extend your understanding through an optional challenge. The test will be on Thursday, November 17. On Friday, November 18th we will have our annual casino day. We would appreciate adult help on this day, especially from parents who have some experience watching chips pass back to the "house." If you want to design a casino game of chance where you will be the "house" and the students will play against you, see Mrs. L this week.

Please be safe on Tuesday. Good luck to the GHP interviewees. See you all on Wednesday.

Chapter 5 - Experimental design, sampling, simulation

2005-10-30T16:58:00.000-05:00

The test wil be Monday, November 8. Be thinking about what experiment or data collection activity you can perform during your lunch on Wednesday at the honor card event.