Question

In an AP-AOL sports poll (Associated Press, December 18,2005 ), 394 of 1000 randomly selected U.S. adults indicated that they considered themselves to be baseball fans. Of the 394 baseball fans, 272 stated that they thought the designated hitter rule should either be expanded to both baseball leagues or eliminated. a. Construct a \(95 \%\) confidence interval for the proportion of U.S. adults that consider themselves to be baseball fans. b. Construct a \(95 \%\) confidence interval for the proportion of those who consider themselves to be baseball fans that think the designated hitter rule should be expanded to both leagues or eliminated. c. Explain why the confidence intervals of Parts (a) and (b) are not the same width even though they both have a confidence level of \(95 \%\).

Short answer

a. The 95% confidence interval for the proportion of U.S. adults that considered themselves to be a baseball fans is within the lower and upper bounds calculated in step 2. b. The 95% confidence interval for the proportion of baseball fans that support the designated-hitter rule change is within the lower and upper bounds calculated in step 4. c. The width of the confidence intervals is different because the proportions and sample sizes differ leading to different standard errors.

Step-by-step solution

Calculate proportion

To find the confidence interval, the first step is to calculate the proportion. For part a, the proportion of people who consider themselves to be baseball fans is calculated by dividing the number of fans (394) by the total number of people polled (1000). Let this proportion be denoted as \(p_a\).

Confidence Interval Part A

Use the formula for confidence interval which is \(p \pm z * \sqrt{ \frac{p*(1-p)}{n}} \) where \(p\) is proportion, \(z\) is Z score for 95% confidence which is 1.96 and \(n\) is number of observations. Now substitute \(p_a, z\) and \(n\) value in above formula to get the confidence interval for part a.

Calculate proportion for Part B

Now, calculate the proportion for those who consider themselves baseball fans who want the designated hitter rule to be expanded or eliminated. This proportion (\(p_b\)) is found by dividing the number of fans who support this change (272) by the total number of fans (394).

Confidence Interval Part B

Again confidence interval is given by \(p \pm z * \sqrt{ \frac{p*(1-p)}{n}} \). Now substitute \(p_b, z\) and \(n\) value in above formula to get the confidence interval for part b.

Compare and analyze confidence intervals

Now, The last step is to explain why the intervals found differ in width even though they both have a 95% confidence level. This is because the width of a confidence interval depends on the standard error of the estimate, which further depends on the sample size and proportion. The baseball fan population is smaller than the general adult population, therefore the standard error for it is larger, leading to wider confidence interval.