Points: A Random Pool Of 1200 Loan Applicants Attending Four

1 10 Points A Random Pool Of 1200 Loan Applicants Attending Four

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. The median balance was $1645, indicating a skewed distribution. Assuming the standard deviation for the population is $3500, compute a 95% confidence interval for the true mean credit card balance among all undergraduate loan applicants.

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The objective of this analysis is to estimate the range within which the true average credit card balance for all undergraduate loan applicants lies, based on the sample data. Given the large sample size (n=1200), the Central Limit Theorem assures that the sampling distribution of the sample mean approximates normality, allowing us to use the normal distribution to construct the confidence interval even with a skewed population distribution.

The sample mean (x̄) is reported as $3173, and the population standard deviation (σ) is assumed known as $3500. The sample size (n) is 1200. The confidence level selected is 95%, which corresponds to a z-value of approximately 1.96 for a two-tailed confidence interval.

The formula for the confidence interval when the population standard deviation is known is:

CI = x̄ ± z*(σ/√n)

Calculating the standard error (SE):

SE = σ / √n = 3500 / √1200 ≈ 3500 / 34.64 ≈ 101.07

Calculating the margin of error (ME):

ME = z SE = 1.96 101.07 ≈ 198

Constructing the confidence interval:

  • Lower bound: 3173 - 198 = 2975
  • Upper bound: 3173 + 198 = 3371

Therefore, the 95% confidence interval for the true mean credit card balance among all undergraduate loan applicants is approximately ($2975, $3371). This interval suggests that, with 95% confidence, the true average credit card balance for the population falls within this range.

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