Need help with all parts, please

Question 1. We want to gure out what proportion of people in Dutchess County, NY (population: 300,000) have watched The Great British Baking Show. You know the proportion might vary by age so three age groups are considered: Generation A (30% of the population); Generation B (20% of the population); and Generation C (50% of the population). There are two options for the survey: (1) Random sample: 100 people are chosen uniformly at random from the whole county. X is the number of those 100 who answer Yes. (2) Stratied sample: we sample uniformly within each age group: 30 people from Generation A, 20 people from Generation B, and 50 people from Generation C. Z A is the number from Generation A who say Yes, Z 5 is the number from generation B who say Yes, and Z0 is the number from generation C who say Yes. The overall number who answer Yes is Z = Z A + Z 3 + Z0. Assume that 10A, p3, and 330 are the proportion of the population who have watched the show among Generation A, Generation B, and Generation C, respectively. Let p = 0.3pA + 0.2103 + 0.5300 be the overall proportion who have watched the show. (a) Each of X, Z A, Z 3, and Zc can be modeled using a binomial distribution. Fill in the blanks with the appropriate parameters for these distributions: X m Bin( ,_). Z A N Bm( ,_). Z B N Bin( ,_). ZC ~ Btn( , ). Additionally, explain why we can assume that Z A, Z 3, and Z0 are independent in the stratied sample. (b) Find E[X/100] and E[Z/100] in terms of p. (c) Find Var(X / 100) and Var(Z / 100) in terms of p and (where necessary) 10,4, 103, and pg. ((1) Using the stratied sample, we can also compare age groups. Find the values of E [% gin] and Va (is: , s)- (e) Assume pA = 0.1, p3 = 0.2, and p0 = 0.3, so 39 = 0.22. Find the expectations and standard deviations of X / 100 and Z / 100 using your results from parts (b) and (c). Which of the two survey strategies would you use? Explain your choice to someone who has not taken a probability course. (f) Suppose we mess up on our stratied sample and sample 40 people from Generation A, 30 from Generation B, and 30 from Generation G. Then what will E[Z / 100] and Var(Z / 100) be, in terms of 13, 10A, 103, and pg? What's the problem with this