Question

The article "Students Increasingly Turn to Credit Cards" (San Luis Obispo Tribune, July 21,2006 ) reported that \(37 \%\) of college freshmen and \(48 \%\) of college seniors carry a credit card balance from month to month. Suppose that the reported percentages were based on random samples of 1,000 college freshmen and 1,000 college seniors. (Hint: See Example 9.5\()\) a. Construct and interpret a \(90 \%\) confidence interval for the proportion of college freshmen who carry a credit card balance from month to month. b. Construct and interpret a \(90 \%\) confidence interval for the proportion of college seniors who carry a credit card balance from month to month. c. Explain why the two \(90 \%\) confidence intervals from Parts (a) and (b) are not the same width.

Short answer

The 90% confidence interval for the population proportion of college freshmen who carry a credit card balance is between 34.55% and 39.45%. The 90% confidence interval for the population proportion of college seniors who carry a credit card balance is between 45.56% and 50.44%.

Step-by-step solution

Calculate the Sample Proportions

First, calculate the sample proportions. In the sample of 1,000 college freshmen, 37% carry a balance, so the sample proportion (\(\hat{p_1}\)) is 0.37. In the sample of 1,000 college seniors, 48% carry a balance, so the sample proportion (\(\hat{p_2}\)) is 0.48.

Construct the Confidence Intervals

Now, construct the 90% confidence intervals using the formula: \(\hat{p} \pm z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\), where \(z\) is the z-score for the desired level of confidence. For a 90% confidence interval, \(z = 1.645\). For college freshmen, the CI is: \(0.37 \pm 1.645\sqrt{\frac{0.37(1-0.37)}{1000}}\) and for college seniors, the CI is: \(0.48 \pm 1.645\sqrt{\frac{0.48(1-0.48)}{1000}}\).

Calculate the Confidence Intervals

Next, calculate the actual confidence intervals. For college freshmen: \(0.37 \pm 1.645(0.017) = [0.3455, 0.3945]\) For college seniors: \(0.48 \pm 1.645(0.016) = [0.4556, 0.5044]\).

Interpret the Confidence Intervals

The 90% confidence interval for the population proportion of college freshmen who carry a credit card balance is between 0.3455 and 0.3945. This means that we are 90% confident that the true proportion of college freshmen carrying a credit card balance is within this interval. Similarly, the 90% confidence interval for the population proportion of college seniors who carry a credit card balance is between 0.4556 and 0.5044. We are 90% confident that the true proportion of college seniors carrying a credit card balance is within this interval.

Explain the Differences in Confidence Intervals

The two confidence intervals are not the same width because the sample proportions are different. A bigger sample proportion means a larger variability, resulting in a wider confidence interval. As a result, the confidence interval for college seniors is wider than for college freshmen.