You intend on buying a car. After you pick the make and model that you want, you decide to figure out what it should cost you. You look up 100 different sales of that specific car and find that the average selling price was $27, 686 with a standard deviation of $1,365.

(a) Construct and interpret a 95% confidence interval for the true cost of the car you wish to buy. (7pts)

(b) How large does your sample size need to be in order to construct a 99.7% confidence interval that has a width of $500? (7pts)(Note: The width of a confidence interval is the upper bound of the interval minus the lower bound of the interval.)

(c) You decide to wait a year for the car to no longer be the newest model. Once this happens, you take another sample of 50 cars and find that their average price has dropped to $19, 445 with a standard deviation of $1958. Will a 68% confidence interval have a larger or smaller width than the 95% confidence interval produced in part (A)? (Note: Calculate the interval of the width before you make a decision, and show your work.) (7pts)