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 1000 college freshmen and 1000 college seniors. a. Construct a \(90 \%\) confidence interval for the proportion of college freshmen who carry a credit card balance from month to month. b. Construct 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 intervals for the proportion of college freshmen and seniors who carry a credit card balance from month to month are calculated separately due to differing proportions and equal sample sizes. The varying widths of the intervals are due to these factors.

Step-by-step solution

Calculate standard error for both populations

The standard error for proportion is calculated as \(SE = sqrt[p(1-p)/n]\), where p is the proportion and n is the sample size. For college freshmen, \(SE_1 = sqrt[0.37(1-0.37)/1000]\). For college seniors, \(SE_2 = sqrt[0.48(1-0.48)/1000]\). Respectively, calculate these values.

Find the z-value corresponding to a 90% confidence level

Using a z-table or an online calculator, find the z-value corresponding to a confidence level of 90%. This z-value will be the same for both groups as they both have a 90% confidence level. The z-value for a 90% confidence level is approximately 1.645.

Construct confidence intervals

A confidence interval for a proportion is calculated as \(p ± z*SE\). For freshmen, the confidence interval would be \(0.37 ± 1.645*SE_1\). For seniors, it would be \(0.48 ± 1.645*SE_2\). Calculate these ranges.

Explain the difference in widths

The width of a confidence interval is determined by the product of the z-value and the standard error. Since the z-value is constant for both groups, the difference in width must be due to differences in the standard errors, which in turn are influenced by the different proportions and equal sample sizes for the two groups.