Use the standard normal distribution or the t-distribution to construct a 90% co

Question

Use the standard normal distribution or the t-distribution to construct a 90% confidence interval for the population mean. Justify your decision. If neither distribution can be used, explain why. Interpret the results.

The gas mileages (in miles per gallon) of 45 randomly selected cars are listed below.

24 13 14 26 17 23 26 18 17 19 13 22 17 23 20 20 27 13 21 23 24 17 21 16 13 14 25 22 18 21 21 23 17 21 14 15 19 19 23 26 26 17 23 24 14

Which distribution should be used to construct the confidence interval?

A.Use a t-distribution because ngreater than or equals30 and sigmais known.

B.Use a t-distribution because ngreater than or equals30 and sigmais unknown.

C.Use a normal distribution because ngreater than or equals30 and sigmais known.

D.Use a normal distribution because ngreater than or equals30 and sigmais unknown.

E.Cannot use the standard normal distribution or the t-distribution because sigmais unknown, n less than 30, and the mileages are not normally distributed.

Select the correct choice below and, if necessary, fill in any answer boxes to complete your choice.

A.The 90% confidence interval is (____,____).

(Round to one decimal place as needed.)

B.Neither distribution can be used to construct the confidence interval.

Interpret the results.

A.It can be said that 90% of cars have gas mileage between the bounds of the confidence interval.

B.With 90% confidence, it can be said that the population mean gas mileage is between the bounds of the confidence interval.

C.If a large sample of cars are taken approximately 90% of them will have gas mileage between the bounds of the confidence interval.

D.Neither distribution can be used to construct the confidence interval.

Explanation / Answer

Since n = 45 is the sample size and n > 30, we can use a standard normal distribution instead of the T distribution. Also in this given question, sigma is unknown which is the standard deviation of the population. Hence correct option is D

Confidence interval states that If repeated samples were taken and the 90% confidence interval was computed for each sample, 95% of the intervals would contain the population mean. So, the correct option is C

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