AP Statistics 4.9: Choosing a Two-Sample t-Test

Identify a two-sample t-test for independent groups and define the population mean difference, response variable, and populations.

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.

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4.9.A—Identify an appropriate testing method for the difference between two population means including the parameters… question 1