For the population of N = 5 units of Exercise 3 of Chapter 2:

(a) Compute directly the variance v_ar(y) of the sample mean and the variance v_ar(m) of the sample median.

(b) From each sample, compute the sample variance s² and the estimate v_ar (y) of the variance of the sample mean. Show that the sample variance s² is unbiased for the finite-population variance σ² but that the sample standard  is not unbiased for the population standard deviation 

(c) For each sample, construct a standard 95% confidence interval for the population mean. What is the actual coverage probability for the method with this population and design?

Exercise 3 Consider a small population of N = 5 units, labeled 1, 2, 3, 4, 5, with respective y-values 3, 1, 0, 1, 5. Consider a simple random sampling design with a sample size n = 3. For your convenience, several parts of the following may be combined into a single table. (a) Give the values of the population parameters μ, τ , and σ². List every possible sample of size n = 3. For each sample, what is the probability that it is the one selected? (b) For each sample, compute the sample mean y and the sample median m. Demonstrate that the sample mean is unbiased for the population mean and determine whether the sample median is unbiased for the population median.