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.
Friday, February 19, 2010
Monday, January 11, 2010
The Central Limit Theorem and Sampling Distributions
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.
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.
Tuesday, November 17, 2009
Chapter 7 - Random Variables
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.
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.
Wednesday, November 04, 2009
Chapter 6 - Probability
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.
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.
Tuesday, October 27, 2009
Chapter 5 - Producing data
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.
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.
Thursday, October 01, 2009
Chapter 4 Non-linear relationships
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.
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.
Friday, September 11, 2009
Chapter 3 Exploring Linear Relationships
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.
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.
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