Question

Problem 2. Suppose the population has six units: U={1,2,3,4,5,6} and samples of size 3 could be chosen from this population. For purposes of studying sampling distribution, assume that all population values are known y1 92 , y2 = 108, y3 = 154, y4 = 133, y5 = 190, y6 = 175 We are interested in yu, the population mean. One sampling plan is proposed. Sample, S (1,3,5 {1,4,6 {2,3,6 (2,4,5 P(S) Sample Number 1 0.25 2 0.2 3 0.2 0.35 (a) What is the value of yu? (5 pts) (each 1 pt) (b) Let y be the mean of the sample values. For each sampling plan, find (i) E(J); V() (ii) (iii) Bias(y); MSE( (iv)

Answer 1

(a) yu (b) Sample No. Sample (92 108 154 133 +190 175)/6 142 Value of y {1,3,5 (92 154 190)/3 145.3333 {1,4,6}(92133175)/3 133.3333 {2,3, 6 (108+154 175)/3 145.6667 {2,4,5} (108 + 133190)/3 143.6667 145.3333 0.25 133.3333 0.2 145.6667 0.2143.6667* 0.35 P(S) 0.25 1 2 0.2 0.2 4 0.35 (i) E(g) 142.4167 (ii) E() 145.33332*0.25+133.33332*0.2+145.66672+0.2+143.66672+0.35 - 20303.8 V ()E() (iii) Bias(g)= E(g) Ju 142.4167 142 0.4167 (iv) MSE() V()(Bias(y))2 21.28356(0.4167)2 21.4572 E2()20303.8 (142.4167)2 21.28356 _