The loan manager for State Bank and Trust has claimed that the mean loan balance on outstanding loans at the bank is over $14,500.To test this at a significance level of 0.05,a random sample of n = 100 loan accounts is selected.Assuming that the population standard deviation is known to be $3,000,the value of that corresponds to the critical value is approximately $14,993.50.

Question

The loan manager for State Bank and Trust has claimed that the mean loan balance on outstanding loans at the bank is over $14,500.To test this at a significance level of 0.05,a random sample of n = 100 loan accounts is selected.Assuming that the population standard deviation is known to be $3,000,the value of  that corresponds to the critical value is approximately $14,993.50.

Quizplus · Accepted Answer

This statement is true. The loan manager is claiming that the population mean loan balance is greater than $14,500. We are testing this claim using a one-tailed test at a significance level of 0.05, which corresponds to a z-score cutoff of 1.645. Using the formula: 
z = (x̄ - μ) / (σ / √n) 
where x̄ is the sample mean, μ is the hypothesized population mean, σ is the population standard deviation, and n is the sample size, we can solve for x̄ and get: 
x̄ = μ + z(σ / √n) 
x̄ = $14,500 + (1.645)($3,000 / √100) 
x̄ = $14,993.50 
This is exactly the same as the given value, so the statement is true.

The Loan Manager for State Bank and Trust Has Claimed