Question

The article "Credit Cards and College Students: Who Pays, Who Benefits?" (Journal of College Student Development \([1998]: 50-56\) ) described a study of credit card payment practices of college students. According to the authors of the article, the credit card industry asserts that at most \(50 \%\) of college students carry a credit card balance from month to month. However, the authors of the article report that, in a random sample of 310 college students, 217 carried a balance each month. Does this sample provide sufficient evidence to reject the industry claim?

Short answer

After calculating, we find that the test statistic value is likely to be significantly greater than the critical value. Thus, based on the sample data, we can reject the null hypothesis that no more than 50% of college students carry a credit card balance. It appears likely that more than 50% of college students carry a credit card balance from month to month.

Step-by-step solution

Setting up the Hypotheses

We set up the null hypothesis \(H_0\) and the alternative hypothesis \(H_a\). In this case, the null hypothesis is that at most 50% of students carry credit card debt, or \(p \leq 0.50\), and the alternative hypothesis is that more than 50% of students carry debt, or \(p > 0.50\).

Calculating the Sample Proportion

We calculate the sample proportion \(\hat{p}\) which is equal to the number of 'successes' or students who carry a balance each month divided by the total number of students. In our case, this is \(217/310 \approx 0.6996\).

Calculating the Test Statistic

To conduct hypothesis testing, we need to standardize our sample statistic using the formula for the test statistic which is \((\hat{p} - p_0)/\sqrt{ (p_0*(1-p_0)) / n}\), where \(p_0\) is the claimed population proportion, \(\hat{p}\) is the sample proportion and \(n\) is the sample size. Therefore, the test statistic \(z\) is calculated as \((0.6996 - 0.5)/\sqrt{ (0.5*(1-0.5)) / 310}\).

Determining the Critical Value

We need to determine the critical Z value for the hypothesis test. Because the problem posits a 'greater than' alternative hypothesis, we would use a one-tailed z-test. This would give us a critical Z value, typically \(\pm 1.645\) for a 5% level of significance in a one-tailed test.

Deciding Whether to Reject or Fail to Reject \(H_0\)

We compare the calculated test statistic with the critical value to decide whether to reject or fail to reject the null hypothesis. If the test statistic calculated in Step 3 is greater than the critical value, we reject the null hypothesis. If it is less, we do not reject the null hypothesis.