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.
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12 comments:
10.69 I don't understand why you used 1.96 when the problem says to use 1.645. Are we looking at the same problem? When z = 1.645 you have 5% in that tail. One tail 5% ==> 1.645. Two tail 5%, .025 in each tail==>1.96.
For 10.69 you do what we did in class today. You can look at the answer in the back of the book to confirm your answers.
Step 1: write the rule. Write down the conditions for rejecting the null hypothesis. When will you reject the null for Ho: mu = 0, Ha: mu > 0? When x-bar is larger than some cut-off value. Figure out what that value is.
Step 2: If we throw out the original distribution in favor of mu = 1.1, what portion of the time would we have rejected? That is the POWER. What portion of the time would we have failed to reject? That is BETA, the probability of a type II error.
Ci you there.
Ha ha.
The authors already standardized the results. When they said "The one-sample t-statistic. . .has the value t = 1.12" they gave you the result. Now, compare that t-value to the values on the correct line of the t-table. Where would 1.12 fall? How much (total) would be in the two tails, approximately? You can use tcdf on your calculator to verify these findings.
Look at Step 3 of Example 11.2 for more info.
I hadn't planned to do CiCi's this weekend, since I'm still trying to get better. How many of you guys are planning to do CiCi's?
Jenny, you (and a few more) are coming in Tuesday morning to take the make-up test! 7:20 in room 229!
Don't forget.
ALPHA, POWER, AND BETA
When you perform a hypothesis test for mu you can choose an alpha-level or you can just evaluate your p-value as a likelihood. For instance, let's say that the p-value you calculated was .07. You could say that because .07 is greater than our alpha level of .05, we fail to reject the null hypothesis [add more contextual stuff here].
OR
You could say that while there is some evidence to suggest that the null hypothesis is incorrect when the p-value = .07, the evidence is not extremely strong. Therefore we might choose to reject or fail to reject, based on how sensitive we want to be.
Now, if we established a value for alpha early on, we determined what portion of the time we would be willing to reject the null hyp. when it was actually true. For instance, if we chose 5%, then we're saying that if our x-bar falls out in the 5% in the extreme part of the tail--and it WILL 5% of the time!--then we will reject, even though that is actually a mistake. The point is that we don't know if the null is right or wrong, so we do the best we can.
So, alpha is the probability that we unknowingly reject the null when we should not have, that we made a type I error. We get to choose the likelihood of this.
But, what if. . .
what if some other value is the actual value of mu? Then we made a mistake if we failed to reject and we responded correctly if we rejected the null hypothesis. The likelihood that we did the right thing is the power. The probability that we did the wrong thing (made a type II error) is beta. Both of these probabilities are calculated using the NEW, IMPROVED PROBABILITY CURVE ceneterd at the NEW, IMPROVED mu.
Figure out where you would have rejected and failed to reject under the original rules (the null hyp). Then find the area above the rejection region to get the power and the area above the FTR region to get the beta. Since these two (or three) regions make up all of the area under the probability curve, they add up to 1.
In summary, 1-beta = power.
1-power = beta
Alpha is measured under a different curve, so you can't add it to these numbers to get anything useful.
Now that you know where these numbers come from, what effect will increasing alpha, n, or sigma have on the power or the likelihood of a type II error? Don't just concentrate on answering other people's questions. . .generate your own deep questions and answer them!!!
C'mon, guys! Use your resources. The Barron's guide has a whole page on this stuff.
My summary--and you'll still have to do footwork:
Check the assumptions or conditions that let you
(1) use this test to make an inference about the population [basically, is it an SRS?],
(2) use the simplified form of the sampling distribution standard deviation,
(3) use a normal distribution to model your sampling distribution,
and any other assumptions/conditions that may be necessary because you recognize that your data have some other form, like binomial or geometric.
It looks like you guys haven't recognized when the rules of thumb are used and when they are inappropriate. I've answered this question before. HINT: look at the section headings as you READ THE TEXT.
For the first question, look at the part that says, "The project hoped to show. . ." This is where they got the less than part. They must be saying that x = time with right hand minus time with left hand.
Now, for the second part. In your reading from the weekend you learned that you find the difference of the two values for each participant and use that list of results as your list of x values. Therefore, there is only one mean and one standard deviation for your sample.
According to my quick and dirty subtraction, the numbers should be -24, 0, -3, -7, 23, etc.
For problem 16:
Why did you set the null hypothesis mu so high? The bank is trying to see if the amount charged increases. What should the hypothesis be?
For part b, you have the x-bar, the n, and the sample standard deviation. What more do you need? I think you can change the setting on the calculator to STATS.
Well, that's refreshing. What do you know about the difference between the strength of right and left hands?
Guys,
This cough is killing me. Can I trust you to answer each others' questions?
Hack, hack.
Mrs. L
INV-T
The "quick" way to get t-star from your calculator is to use t-int, with x-bar equal to 0, s = sqrt n, and n = n. This gives you the critical t-star values for a specific C.
This works because the center will be zero and the value of s/sqrt n = sqrt n/sqrt n, which equals 1. Thus, the two answers are -t-star and t-star.
Jenny, I think that this is the situation your friend was addressing.
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