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A random pool of 1200 loan applicants attending four-year public or private colleges and universities was analyzed. The sample of applicants carried an average credit card balance of $3173, with a median balance of $1645, indicating a skewed distribution. Assuming the population standard deviation is $3500, compute a 95% confidence interval for the true mean credit card balance among all undergraduate loan applicants.

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Understanding the credit card debt among undergraduate loan applicants is essential for financial institutions, policymakers, and educational administrators. Accurate estimation of the population mean credit card balance offers insights into borrowers’ financial behaviors and helps design targeted financial assistance or regulation. In this paper, we compute a 95% confidence interval for the mean credit card balance of all undergraduates attending four-year colleges, utilizing sample data and the properties of the normal distribution.

Given data indicates a sample mean (\(\bar{x}\)) of $3173, a known population standard deviation (\(\sigma\)) of $3500, and a sample size (n) of 1200. The median balance of $1645 shows the distribution's skewness; however, the large sample size allows us to invoke the Central Limit Theorem, justifying the assumption of normality in the sampling distribution of the mean.

To compute the confidence interval, we use the formula:

CI = \(\bar{x} \pm z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}\)

Where \(z_{\alpha/2}\) is the z-value corresponding to the desired confidence level (95%). For a 95% confidence interval, \(\alpha=0.05\), and \(z_{0.025} \approx 1.96\).

Plugging in the values:

  • Sample mean, \(\bar{x} = 3173\)
  • Population standard deviation, \(\sigma = 3500\)
  • Sample size, \(n=1200\)
  • Z-value for 95% confidence, \(z_{0.025} = 1.96\)

The standard error (SE) is:

SE = \(\frac{\sigma}{\sqrt{n}} = \frac{3500}{\sqrt{1200}} \approx \frac{3500}{34.641} \approx 101.0\)

Therefore, the confidence interval is:

CI = 3173 \(\pm 1.96 \times 101.0\) ≈ 3173 \(\pm 198\)

This results in a confidence interval ranging from approximately $2975 to $3371.

Interpretatively, we can be 95% confident that the true mean credit card balance of the population of undergraduate students attending four-year colleges falls between $2975 and $3371. This interval captures the central tendency of credit card debt among the sampled population, accounting for sampling variability.

This method hinges on the assumption that the population standard deviation is known. While the sample median suggests skewness, the large sample size justifies the use of the normal approximation due to the Central Limit Theorem, which alleviates concerns about distribution skewness affecting the accuracy of the confidence interval.

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