Question #1 (both 454 & 854 - 10 points): Recall that some sampling designs p(S) have a fixed sample size n (e.g., SRS), whereas others have a random sample size ns (e.g., Bernoulli sampling). Prove that 1 V(t) = 2 ( - ) ( 2 kU LEU is equivalent to V() = ( - ) " kU LEU for fixed size sampling designs. Note that you are not allowed to assume any given sampling design (this includes SRS). Question #2 (both 454 & 854 - 5 points): Let's consider a very small finite population U consisting of the following N = 10 units: Y = 125 Y6 136 y2 = 140 y7 = 158 Y3 = 147 98 = 125 94 = 120 Y9 = 161 95 = 166 310 = 160 From that population, we draw a random sample of size n = 3 using either sampling design I (given in table 1) (a) compute E(Y) (b) compute V(y) (c) compute Bias() Sample # S P(S) I1 {1,3,5} 2/9 12 {5,7,9} 1/9 {2,4,6} 1/9 14 {4, 8, 10} 1/9 15 {6,8,9} 2/9 16 {5, 8, 10} 2/9 Table 1: Sampling design I (6 possible samples may be chosen). Hint: Though you can use V() = ( ) KEU LEU to answer part (b), this approach is not the fastest nor the simplest. Assign #1 - STAT 454/854 (W2024) 2 of 8