AP Statistics 4.9 Setting Up a Test for the Difference Between Two Population Means Questions

Set up a two-sample t-test by confirming independent quantitative groups, matching hypotheses to the ordered mean difference, and checking conditions.

Exam points

choose a two-sample t-test for independent quantitative groups and distinguish it from paired analysis
set H0 as equal population means and match the ordered one- or two-sided alternative to the claim
check randomization and independence between groups before making population-level inference
with small samples, inspect both group distributions for strong skewness, outliers, and normality

Covered syllabus points

4.9.A—Identify an appropriate testing method for the difference between two population means including the parameters…4.9.B—Identify the null and alternative hypotheses for the difference between two population means 4.9.C—Justify the appropriateness of a hypothesis test for the difference between two population means by verifying…

Stefan, a psychologist, conducted a study to investigate the effect of time of day on reading comprehension in children. One hundred children volunteered, with their parents' consent, to participate in the study. Fifty of the children were randomly assigned to read a story at 9 a.m.

and then answer 25 questions about it. The remaining 50 children were assigned to read the same story at 3 p.m. and answer the same 25 questions. The reading comprehension for each child was measured by a reading score, which was determined by the number of questions that were answered correctly about the story. Stefan is interested in comparing the mean reading scores for the two times of day. Table 1 shows the results of Stefan's study.

Table 1: Summary Statistics of Reading Scores

Stefan found the conditions for inference were met and conducted a two-sample t-test for the difference in two population means. Let $\mu_{\mathrm{AM}}$ represent the mean reading score for all children, similar to those in the study, who would read the story at 9 a.m. Let $\mu_{\mathrm{PM}}$ represent the mean reading score for all children, similar to those in the study, who would read the story at 3 p.m.

Stefan's hypotheses are as shown.

\begin{aligned} & \mathrm{H}_{0}: \mu_{\mathrm{AM}}=\mu_{\mathrm{PM}} \\ & \mathrm{H}_{\mathrm{a}}: \mu_{\mathrm{AM}} \neq \mu_{\mathrm{PM}} \end{aligned}

Explain why it was appropriate for Stefan to conduct a two-sample t-test for the difference in two population means instead of a paired t-test for the population mean difference.

	Model Solution	Scoring
B	It was appropriate for Stefan to conduct a two-sample t-test instead of a paired t-test because the two groups are independent. Stefan used random assignment to place the 100 volunteer children into two groups, and there is no indication that the two groups of 50 children are paired in any meaningful way (e.g., age, reading comprehension level).	Essentially correct (E) if the response satisfies the following two components: 1. States that the two-sample t-test is appropriate because the groups of children are independent OR that the groups of children were not paired in a meaningful way 2. Explains why the groups are independent instead of paired, either by referencing the use of random assignment of the 100 children to the two groups by providing an explanation that clearly separates the two situations, such as an example of how the data could have been paired Partially correct (P) if the response satisfies only one of the two components required for E if the response states that a two-sample t-test is appropriate because different samples were used but does not explicitly state that the children in the two samples were not paired in a meaningful way. Incorrect (I) if the response does not meet the criteria for E or P.

[Maximum number: 1]

A physician believes that the exercise habits of East Coast adults are different from the exercise habits of West Coast adults. To study this, she gathers information on the number of hours of exercise per week from a random sample of East Coast adults and a random sample of West Coast adults. Which of the following might be an appropriate null hypothesis for this study?

The average number of hours of exercise per week for East Coast adults is different from the average number of hours of exercise per week for West Coast adults.

The average number of hours of exercise per week for East Coast adults is the same as the average number of hours of exercise per week for West Coast adults.

The average number of hours of exercise per week for East Coast adults is greater than the average number of hours of exercise per week for West Coast adults.

The average number of hours of exercise per week for East Coast adults is less than the average number of hours of exercise per week for West Coast adults.

The probability is 0.5 that an East Coast adult and a West Coast adult exercise an equal number of hours per week.

[Maximum number: 1]

A college admissions officer is interested in comparing the SAT Math scores of high school applicants who have and have not taken AP Statistics. She randomly pulls the files of five applicants who took AP Statistics and five applicants who did not, and proceeds to run a t-test to compare the mean SAT Math scores of the two groups. Which of the following is a necessary assumption?

The population variances from each group are known.

The population variances from the two groups are equal.

The population of all SAT scores from each group is roughly normally distributed.

The samples must be independent simple random samples, and for each sample, n p and n(1-p) must both be at least 10 .

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Covered syllabus points

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