Using R or R Markdown for calculations: Suppose is the average height of a college male. You measure the heights (in inches) of twenty college men, getting data x1, , x20, with sample mean x = 68.55 in. and sample variance s^2 = 16.24 in^2 . Suppose that the xi are drawn from a normal distribution with unknown mean and unknown variance ^2 .

(a) Using the sample mean and variance, construct a 90% confidence interval for .

(b) Now suppose you are told that the height of a college male is normally distributed with standard deviation 3.27 in. Construct a 90% confidence interval for .

(c) In part (b), how many people in total would you need to measure to bring the width of the 90% confidence interval down to 1 inch?

(d) Consider again the case of unknown variance in (a). Based on this sample variance of 16.24 in2, how many people in total should you expect to need to measure to bring the width of the 90% confidence interval down to 1 inch? Is it guaranteed that this number will be sufficient? Explain your reasoning.