Question

A credit bureau analysis of undergraduate students credit records found that the average number of credit cards in an undergraduate's wallet was 4.09 ("Undergraduate Students and Credit Cards in 2004," Nellie Mae, May 2005 ). It was also reported that in a random sample of 132 undergraduates, the sample mean number of credit cards that the students said they carried was 2.6 . The sample standard deviation was not reported, but for purposes of this exercise, suppose that it was 1.2 . Is there convincing evidence that the mean number of credit cards that undergraduates report carrying is less than the credit bureau's figure of \(4.09 ?\)

Short answer

To find the final answer, one needs to perform a Hypothesis Test using the calculated Z score and find the corresponding p-value. If the p-value < 0.05, this means that the number of credit cards students carry is statistically significantly less than 4.09.

Step-by-step solution

Set Up Hypotheses

The null hypothesis \(H_0\) would be that the mean number of credit cards that undergraduates carry is equal to 4.09. This can be mathematically represented as \(H_0: \mu = 4.09\). The alternative hypothesis \(H_1\) would then be that the mean number of credit cards is less than 4.09, represented as \(H_1: \mu < 4.09\).

Calculate Standard Error

The standard error (SE) can be calculated using the formula for the standard deviation (\(s\)) divided by the square root of the sample size (\(n\)). In this case, \(SE = \frac{s}{\sqrt{n}} = \frac{1.2}{\sqrt{132}}\).

Find the Test Statistic

The test statistic is calculated using the formula \(Z = \frac{\bar{x} - \mu_0}{SE}\), where \(\bar{x}\) is the sample mean, \(\mu_0\) is the population mean, and SE is the standard error calculated in Step 2. This gives \(Z = \frac{2.6 - 4.09}{SE}\).

Interpret the Result

With this computed Z-score, find the corresponding p-value from the Z-table. If the p-value is less than the significance level (usually 0.05), then there is convincing evidence to reject the null hypothesis and therefore, there's a strong evidence that the students carry less than 4.09 credit cards on average.