Let X,..., Xn be i.i.d. X, where X is the SAT score of a high school senior in Chicago. A researcher is interested in 0, the fraction of high school seniors in Chicago with a SAT score higher than 1200. (a) Write as the expected value of a suitable random variable. (b) Propose an estimator, n, of 0 that is unbiased and consistent. Prove unbiasedness and consistency. (c) What is Var[n]? (d) Propose a consistent estimator for nVar[n]. Justify your answer. (e) A researcher wishes to test the null hypothesis that at least quarter of high school seniors in Chicago scored higher than 1200 in SAT at significance level a. In what follows, suppose that it is known that 0 < P{X > 1200} < 1. i. Formally state the null and alternative hypotheses. ii. Suppose your sample size is large. How would you perform the test? Write down your test statistic, critical value, and the rule you would use to determine whether or not to reject the null hypothesis. iii. State in words the definition of p-value. What is p-value for your test?