Question

In an AP-AOL sports poll (Associated Press, December 18,2005\(), 394\) of 1,000 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 and interpret a \(95 \%\) confidence interval for the proportion of U.S. adults who consider themselves to be baseball fans. b. Construct and interpret a \(95 \%\) confidence interval for the proportion of baseball fans who 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 who are baseball fans is, approximately, \(0.394 \pm 0.030\). b. The 95% confidence interval for the proportion of baseball fans who supports the designated hitter rule change is approximately \(0.690355 \pm 0.047\). c. The widths of the confidence intervals are different despite the same confidence level because of the different sample sizes used in Part (a) and Part (b).

Step-by-step solution

Building 95% Confidence Interval for U.S Adult Baseball Fans

A 95% confidence interval for a proportion can be constructed using the formula: \[ p \pm Z \cdot \sqrt{{p(1-p) \over n}} \] where \(p\) is the proportion of the sample, \(Z\) is the Z-score (which is 1.96 for a 95% confidence interval), and \(n\) is the sample size. Here, \(p = 0.394\) (proportion of adults who are baseball fans) , \(n = 1000\) (total number of adults), and \(Z = 1.96\). By substituting these values into the formula you get: \[0.394 \pm 1.96 \cdot \sqrt{{0.394(1-0.394) \over 1000}}\]

Building 95% Confidence Interval for Baseball Fans supporting Rule Change

Using the same formula to construct the 95% confidence interval for the proportion of baseball fans supporting the rule change. Here, \(p = 0.690355\) (proportion of baseball fans supporting the rule change), \(n = 394\) (total number of baseball fans), and \(Z = 1.96\). Substituting these values into the formula gives: \[0.690355 \pm 1.96 \cdot \sqrt{{0.690355(1-0.690355) \over 394}}\]

Explaining why Confidence Intervals are of Different Widths

The width of a confidence interval is determined by three factors: the confidence level (which was the same for both parts, \(95%\)), the variability in the population (which is reflected here by the variability in the sample), and the size of the sample. In this case, the difference in sample size for part (a) and part (b) is responsible for the difference in width. As the sample size decreases, the uncertainty — and therefore the width of the confidence interval — increases.