Sanderson M. Smith
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Example of MATCHED PAIRS t PROCEDURE
(Problem #5 on 2001 AP Statistics Examination)
This is an analysis of Problem #5 in Section II on the 2001 Advanced Placement Statistics Examination.
The College Board does not allow a reproduction of the problem statement here. If you do not have before you a statement of the actual problem, you can get it from the College Board site
Here is a solution to Problem #5 on the 2001 AP STATISTICS EXAMINATION.
A paired t test of differences is appropriate here.
The ten sample differences (# mg. in name brand - #mg. in generic brand) are as follows:
Here is a dotplot displaying the differences.
Assumptions check: The differences represent a random sample from a population of differences (premise states pharmacies were randomly selected), and the distribution of differences is approximately normal. A modified box plot (TI-83) shows no outliers.
Using TI-83, we find mean of sample differences = 6.6, with s = 5.27. If md =the population mean of sample differences, then the null and alternate hypotheses are
The calculated t statistic is
At the 5% level of significance, the t-distribution table indicates that H0 would be rejected if t > 2.262 or if t < -2.262. Our calculated t statistic of t = 3.96 is in the critical region, so we reject H0 at the 5% significance level. The consumer group should report that the difference (in milligrams) between the amount of active ingredient in the name brand and the amount in the generic brand is significant.
We can further note that the p-value (calculated from TI-83) for t = 3.96 with 9 degrees of freedom is 2*tcdf(3.96,1E99,9) = 0.0033 = 0.33%. In other words, H0 would be rejected at the 1% level of significance, and even at the 1/2% level of significance.
Additional notes relating to this problem:
Two alternate, potentially correct approaches, are shown below. Note that one would have to apply appropriate details and explanations to get complete credit for such solutions.
The following statistical tests would not be appropriate for this problem.