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

12

mortgage​ institutions, the mean interest rate was

3.48%

and the standard deviation was

0.32%.

Assume the interest rates are normally distributed.

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

A.Use a​ t-distribution because the interest rates are normally distributed and

sigmaσ

is known.

B.Use a normal distribution because

nless than<30

and the interest rates are normally distributed.

C.Use a​ t-distribution because it is a random​ sample,

sigmaσ

is​ unknown, and the interest rates are normally distributed.

D.Use a normal distribution because the interest rates are normally distributed and

sigmaσ

is known.

E.Cannot use the standard normal distribution or the​ t-distribution because

sigmaσ

is​ unknown,

n less than 30n<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.

Interpret the results. Choose the correct answer below.

A.If a large sample of institutions are taken approximately

95%

of them will have an interest rate between the bounds of the confidence interval.

B.With

95%

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

C.It can be said that

95​%

of institutions have an interest rate between the bounds of the confidence interval.

D.

Neither distribution can be used to construct the confidence interval.

Answer 1

a) Correct answer: option (C).Use a​ t-distribution because it is a random​ sample, sigmaσ is​ unknown, and the interest rates are normally distributed.

since population SD is not given and the sample is taken from normal population

If population SD is known then we have to use normal distribution to construct confidence interval

If population SD is not known then we have to use 't' distribution to construct confidence interval

b)

The 95% confidence interval is (3.47, 3.49)

c) Correct answer: option (B).With 95% confidence, it can be said that the population mean interest rate is between the bounds of the confidence interval.