Inference for Proportions

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.