Suppose the city council hires you to analyze citizens' satisfaction rate of a policy. You decide to conduct a survey which asks the citizens how they feel about the policy. The survey is coded into one of the three values: -1, 0, or 1, corresponding to "negative", "neutral", or "positive", respectively. You assume that for the population, a proportion p of the citizens have a negative opinion of the policy, a proportion p 0 of the citizens have a neutral opinion of the policy, and a proportion p + of the citizens have a positive opinion of the policy and that p + p 0 + p + = 1. Let {Xi} n i=1 represent the survey responses of individuals in an i.i.d. random sample of size n from the population. You would like to access the satisfaction rate with the sample mean: X = 1 n n i=1 Xi . (a) Find the probability distribution of Xi . (2 marks) (b) Find the expected values and variances of Xi and X, respectively. (4 marks) (c) Can you infer p , p 0 and p + with X? Why or why not? (2 marks) Instead of focusing on X, you decide to consider three estimators: p +, p 0 and p , p + = n i=1 1(Xi = 1) n , p 0 = n i=1 1(Xi = 0) n , p = n i=1 1(Xi = 1) n , where 1(Xi = a) is called identify function such that 1(Xi = a) = 1 when an observation Xi has a value a and 1(Xi = a) = 0 otherwise. In words, p + counts the portion of observations which have positive response in the sample, p 0 counts the portion of observations which have neutral response in the sample, and p counts the portion of observations which have negative response in the sample. (d) Find the expected values and variances of p +, p 0 and p . (4 marks) (e) (Hard) Find the covariance between p + and p . (2 marks) Hint: Carefully write down the covariance formula and expand it by each observation. Can one observation take different values at the same time? Can two independent observations take different values at the same time? (f) Can you infer p , p 0 and p + with p + and p ? Why or why not? (2 marks) (g) Find the (approximate) distribution of p +. (2 marks) (h) Do we know the true variance of p +? Why or why not? (2 marks) You conduct the survey and get the following result. Response Negative Neutral Positive # of obs. 400 300 300 1 For the following parts, you may use this formula to estimate the variance of the sample mean for any random variable Yi : Var\( Y ) = 1 n n i=1 ( Yi Y )2 n . (i) Use X to construct the 95% confidence interval of the expected value of X you found in (b) (assume CLT holds here). (3 marks) (j) Use p and p + to construct the 95% confidence interval of the expected values of p and p + you found in (d) (assume CLT holds here). (4 marks) (k) How would you interpret the results in (i) and (j)? Explain what a confidence interval means. Then compare the two results you solved and give your interpretation. (3 marks)