Please show all steps and explain how to solve these questions.

1. You have been hired to predict the outcome of an election. There are N voters and four candidates: the candidates are labeled A, B, C, D. If you asked each of N voters who they intended to vote for, you would learn that 30% would vote for A 40% would vote for B 10% would vote for C 20% would vote for D. But it is too costly to ask each of the N voters. So, you randomly sample a subset, n, of the voters and ask them who they intended to vote for. Let K be the response from the ith sampled voter and let yz- denote the outcome of Y}: 1 if 73th voter votes for A, 2 if 13th voter votes for B, yd 3 if 73th voter votes for C, 4 if 73th voter votes for D. (a) Compute lE(Y,). Can you learn any PO\"; = y,) directly from EGG)? (3pt) (b) Construct an estimator of the proportion of voters who support A with a sample of n individuals. To construct the estimator, dene a random variable for each event. Show that your estimator is unbiased. (3pt) (c) Derive the formula of the variance for your estimator in (b). When deriving the formula, justify each step. (3pt) Hint: For binary random variable with 0 and 1 as an outcome, the variance is p- (1 p) where p is the probability that the random variable is equal to 1. (d) Now you want to use the regression method learned in this class. Let X,- be equal to 1 if voter 2' votes for A and 0 otherwise. If you run regression with X,- on constant, what will be the coefcient of the constant in this regression? In other words, what is 6 in the following regression equation Xi=6'1+fi? Based on ,8, what is the estimator for a? Is 3 different from the estimator you propose in (b)? (3pt) (e) Supppose that you sample 10 voters and the below table are the answers from them where y,- is dened as in the beginning of this problem: Voters 3/8 299 910 III-III..- For instance, according to this table, the rst voter votes for A whereas the second voter votes for C. What is the sample variance of 3'? Construct the 95% condence interval for ,8. (3pt)