For purposes of studying sampling distribution, we consider a small population of N = 4 units, labeled 1,2, 3, 4, with respective y-values y1 = 3, y2 = 1, y3 = 0, y4 = 5.

(a) Give the values of the population parameters , and 2. (2 marks)

(b) Plan 1: Consider a simple random sample without replacement (SRS) design with sample size

n = 2.

(i) Find the number of possible SRSs of size n = 2. List every possible sample. For each sample,

what is the probability that it is the one selected? (4 marks)

(ii) For each sample, determine y, s2. (4 marks)

(iii) Find E(y), V(y), Bias(y) and MSE(y). (2 marks)

(iv) Find E(s2), V(s2). (1 mark)

(v) Are y and s2 unbiased estimators for and 2? (2 marks)

(c) Plan 2: Consider a simple random sample with replacement (SRSWR) design with sample size

n = 2.

(i) Find the number of possible SRSs of size n = 2. List every possible sample. For each sample,

what is the probability that it is the one selected? (4 marks)

(ii) For each sample, determine y, s2. (4 marks)

(iii) Find E(y), V(y), Bias(y) and MSE(y). (2 marks)

(iv) Find E(s2), V(s2). (1 mark)

(v) Are y and s2 unbiased estimators for and 2? (2 marks)

(d) Which of the two sampling plans in (b) and (c) do you think is better for estimating ? Why?