2. A company wishes to make inference for the proportion of individuals in the United States that use the Google Chrome web browser. They plan to take a survey. Let W1, . . . , W11 be the yetto-be observed survey responses from 7?. individuals, where the event W1- : 1 corresponds to the 2' th individual uses the Google Chrome web browser and the event W1 2 0 corresponds to the i th individual does not use the Google Chrome web browser (3 = 1, . . . ,n). Assume that W1, . . . , W11 are iid Bern(6), where 6 E (0,1). (a) (3 points) What is the distribution of the random variable 221:1 W1 ? This is the yettobe observed number of individuals that use the Google Chrome web browser. Specify the distribution name and the values of its parameter(s) in terms of 6 and n. Then give formulas for E (221:1 W1) and var (2:121 W1) These formulas should involve 6 and n (b) (5 points) Perform a simulation study in R to conrm the formulas in part 2a when n = 10 and 6 = 0.6. To do this, independently repeat the experiment that generates a realization of W1, . . . , W11 and the corresponding realization of 2:121 W1 a total of reps = 10000 times. Report the observed sample mean of the reps realizations of 2:121 W1- as a simulation-based estimate of E (2:1 W1). Report the observed sample variance of the reps realizations of 221:1 W1 as a simulation-based estimate of var (221:1 W1). Are these simulation- based estimates close to their corresponding values from the formulas? (c) (3 points) Let W 2 art1 2:121 W, be the (random) sample mean. Derive formulas for E(W) and var(W). These formulas should involve 6 and n. (d) (3 points) Is W a discrete random variable? Explain. (e) (5 points) Perform a simulation study in R to conrm the formulas derived in part 2 c when n = 10 and 6 = 0.6. To do this, independently repeat the experiment that generates a realization of W1, . . . , W11 and the corresponding realization of W a total of reps = 10000 times. Report the observed sample mean of the reps realizations of W as a simulation-based estimate of E (W) Report the observed sample variance of the reps realizations of W as a simulation- based estimate of var(W). Are these simulationbased estimates close to their corresponding values from the formulas? (f) (5 points) Someone claims that the random interval Wj: 1/ V171, captures 6 with probability approximately equal to 0.95. Create a simulation study in R to test this claim when 9 = 0.6. To do this, independently repeat the experiment that generates a realization of W1, . . . , Wu and the corresponding realization of the random interval W :I: l/ a total of reps = 10000 times. Then report the observed sample proportion of the reps realizations of this random interval that captured 9 = 0.6. Perform this simulation study when n = 35,40, 55, 100, 200. Based on the simulation results, is the coverage probability of this random interval approximately equal to 0.95 for these values of n, and 9