At a certain high school, 38% of freshmen and 31% of sophomores walk to school. Suppose that random samples of 40 freshmen and 45 sophomores are chosen. Let S = the proportion of sophomores who walk to school and F = the proportion of freshmen who walk to school. What is the probability that the proportion of sophomores who walk to school is greater than the proportion of freshmen?Find the z-table here.
The sampling distribution of S − F has mean 0.31 − 0.38 = −0.07 and standard deviation √[(0.31)(0.69)/45 + (0.38)(0.62)/40] ≈ 0.103. The z-score for S − F = 0 is [0 − (−0.07)]/0.103 ≈ 0.679. Thus, P(S > F) = P(Z > 0.679) ≈ 1 − 0.752 = 0.248, choice A.
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