Question

Use the standard normal distribution or the​ t-distribution to construct a 95​% confidence interval for the population mean. Justify your decision. If neither distribution can be​ used, explain why. Interpret the results. In a random sample of 18 mortgage​ institutions, the mean interest rate was 3.49​% and the standard deviation was 0.48​%. Assume the interest rates are normally distributed. Which distribution should be used to construct the confidence​ interval? A. Use a normal distribution because nless than30 and the interest rates are normally distributed. B. Use a​ t-distribution because it is a random​ sample, sigma is​ unknown, and the interest rates are normally distributed. Your answer is correct.C. Use a normal distribution because the interest rates are normally distributed and sigma is known. D. Use a​ t-distribution because the interest rates are normally distributed and sigma is known. E. Cannot use the standard normal distribution or the​ t-distribution because sigma is​ unknown, n less than 30​, and the interest rates are not normally distributed. Select the correct choice below​ and, if​ necessary, fill in any answer boxes to complete your choice. A. The 95​% confidence interval is ​( nothing​, nothing​). ​(Round to two decimal places as​ needed.) B. Neither distribution can be used to construct the confidence interval

Answer 1

Solution:

Given:

Sample size = n = 18

Sample mean =   = 3.49%

Sample standard deviation = s = 0.48​%.

Confidence level = c = 95%

The interest rates are normally distributed.

Which distribution should be used to construct the confidence​ interval?

B. Use a​ t-distribution because it is a random ​sample, sigma is​ unknown, and the interest rates are normally distributed.

A. The 95​% confidence interval is:

where

t_c is t critical value for c = 95%  confidence level

Thus two tail area = 1 - c = 1 - 0.95 = 0.05

df = n - 1 =  18 - 1 = 17

Look in  t table for df = 17 and two tail area = 0.05 and find t critical value

t_c = 2.110

Thus

Thus

Interpretation:

With 95% confidence, it can be said that the population mean interest rate is between the bounds of the confidence interval.