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Explain why we determine a pooled estimate of the population proportion when testing hypotheses regarding the difference of two proportions, but do not pool when constructing confidence intervals about the difference of two proportions.

Suppose a researcher believes the mean from population 1 is less than the mean from population 2 in matchedpairs data. How would you define \(\mu_{d} ?\) How would you determine \(d_{i} ?\)

Determine whether the sampling is dependent or independent. An educator wants to determine whether a new curriculum significantly improves standardized test scores for third-grade students. She randomly divides 80 third-graders into two groups. Group 1 is taught using the new curriculum, while group 2 is taught using the traditional curriculum. At the end of the school year, both groups are given the standardized test and the mean scores are compared.

Does the Designated Hitter Help? In baseball, the American League allows a designated hitter (DH) to bat for the pitcher, who is typically a weak hitter. In the National League, the pitcher must bat. The common belief is that this results in American League teams scoring more runs. In interleague play, when American League teams visit National League teams, the American League pitcher must bat. So, if the DH does result in more runs, we would expect that American League teams will score fewer runs when visiting National League parks. To test this claim, a random sample of runs scored by American League teams with and without their DH is given in the following table. Does the designated hitter results in more runs scored at the \(\alpha=0.05\) level of significance? Note: \(\bar{x}_{\mathrm{NL}}=4.3, s_{\mathrm{NL}}=2.6, \bar{x}_{\mathrm{AL}}=6.0, s_{\mathrm{AL}}=3.5\)

Aluminum Bottles The aluminum bottle, first introduced in 1991 by CCL Container for mainly personal and household items such as lotions, has become popular with beverage manufacturers. Besides being lightweight and requiring less packaging, the aluminum bottle is reported to cool faster and stay cold longer than typical glass bottles. A small brewery tests this claim and obtains the following information regarding the time (in minutes) required to chill a bottle of beer from room temperature \(\left(75^{\circ} \mathrm{F}\right)\) to serving temperature \(\left(45^{\circ} \mathrm{F}\right) .\) Construct and interpret a \(90 \%\) confidence interval for the mean difference in cooling time for clear glass versus aluminum.

A physical therapist wants to determine the difference in the proportion of men and women who participate in regular sustained physical activity. What sample size should be obtained if she wishes the estimate to be within three percentage points with \(95 \%\) confidence, assuming that (a) she uses the 1998 estimates of \(21.9 \%\) male and \(19.7 \%\) female from the U.S. National Center for Chronic Disease Prevention and Health Promotion? (b) she does not use any prior estimates?