A bank examiner is interested in estimating the mean outstanding defaulted loans balance of all defaulted loans over the last three years.A random sample of 20 defaulted loans yielded a mean of $67,918 with a standard deviation of $16,552.40.If the outstanding balance is normally distributed,the 90% confidence interval for the mean balance of defaulted loans over the past three years would be _____.
A)[66,487 69,349]
B)[39,299 96,537]
C)[57,329 78,507]
D)[61,829 74,007]
E)[61,519 74,317]

Question

Quizplus · Accepted Answer

Since the sample size (n=20) is sufficiently large and the population standard deviation is unknown, we will use a t-distribution to construct the confidence interval.

Using a t-distribution with 19 degrees of freedom (df=n-1) and a 90% confidence level, we can find the t-value from the t-table or calculator to be 1.729.

The margin of error (ME) can be calculated using the formula:
ME = t-value * (standard error)
where standard error (SE) = (sample standard deviation / square root of sample size)

ME = 1.729 * (16,552.4 / sqrt(20)) = 7,885.86

To construct the confidence interval, we can use the formula:
(sample mean) ± (ME)

CI = 67,918 ± 7,885.86

CI = (61,032.14, 74,803.86)

Therefore, the 90% confidence interval for the mean balance of defaulted loans over the past three years is (61,519, 74,317).

Thus, the correct choice is E.

A Bank Examiner Is Interested in Estimating the Mean Outstanding