I’m working on a probability multi-part question and need an explanation to help me study.
1.Suppose that x1, x2, x3, x4, x5, is a sample drawn from the exponential distribution f(x; θ) := ( θe−θx if x ≥ 0, 0 otherwise, where θ is an unknown parameter.
(a) Find an expression for the joint likelihood function f(x1, x2, x3, x4, x5; θ).
(b) Is x-bar a sufficient statistic for θ? Explain your answer.
(c) Now suppose that x1 = 5, x2 = 3, x3 = 4, x4 = 2, x5 = 2. Find the maximum likelihood estimate of θ.
2.Let X1, . . . , X120 be a collection of i.i.d. random variables and let Y1, . . . , Y120 be the associated order statistics.
(a) Find a value of j such that the interval [Y30−j, Y30+j ] is an approximate 95% confidence interval for the 1/4 quartile of the distribution.
3.A political candidate hires a polling company to determine their level of support, i.e. the probability p that a random person will vote for them. The polling company will contact n people and collect the data x1, . . . , xn where xi = 1 if person i supports the candidate and xi = 0 if the person does not support the candidate. Thus, the people contacted by the polling company can be treated as i.i.d. b(1, p) random variables. Let ¯x be the sample mean of the data.
(a) How many people does the polling company need to contact in order to guarantee that (x-bar − 0.02, x-bar +0 .02) is a 95% confidence interval for p? (Use the overestimate p(1 − p) ≤ 1/4 to help estimate the variance).
(b)The client believes that their true level of support is ˆp = 0.7. How many fewer people need to be contacted (compared to part (a)) if you instead estimate the variance by replacing p(1 − p) with ˆp(1 − pˆ) = .21 instead?
4.Suppose you have a normal distribution N(µ, 30) where µ is unknown. You want to test the null hypothesis µ = 4 against the alternative hypothesis µ < 4. Suppose you create testing where you collect n data points x1, . . . , xn and reject the null hypothesis if the sample mean x-bar < c for some cutoff value c.
(a) Let K(ν) be the power function of the test at some value ν ∈ R. Give an expression in terms of n, c, ν for K(ν) (note your final expression should either be in terms of the error function erf OR it should be the probability of an N(0, 1) random variable belonging to a certain region).
(b) Find values for c and n that guarantee that the testing has at most 0.005 type 1 error and at most 0.01 type 2 error when µ = 2.
5.A company is testing the efficacy of a new vaccine against a certain virus. Let p0 represent the probability that a random unvaccinated person will contract the virus over the course of two months, and let p1 represent the probability that a random fully vaccinated person will contract the virus over the course of two months. The company will test whether the vaccine is effective by recruiting 20, 000 volunteers and splitting them into two equal groups of size n = 10, 000. One group will receive the vaccine while the other group receives a placebo. After tracking the participants for two months, the company collects the data n0, n1 where n0 is the number of people in the placebo group who contracted the virus and n1 is the number of people in the vaccine group who contracted the virus.
(a) Suppose that n0 = 189. Find a 95% confidence interval for p0 (Note since the sample size is so large if p0 shows up in the variance formula you can replace it with n0 n .)
(b) Let X-bar be a random variable representing the sample mean of the placebo group and let Y-bar be a random variable representing the sample mean of the vaccine group. What is the expected value and variance of W = Y-bar − 1/2 X-bar?
(c) Express the probability that p1 < 1/2 p0 in terms of W.
(d) Suppose that n1 = 11. Using your answers from parts (b) and (c), how confident can the company be that p1 < 1/2 p0? (Again if p0 or p1 show up in a variance formula you can replace them with n0/n and n1/n respectively)
6. Suppose X1, . . . , X60 are i.i.d. b(1, p) random variables where p is unknown. You are given the null hypothesis H0 : p = 1/4.
(a) Suppose you want to test H0 against the simple alternative hypothesis H1 : p = 1/2. Use the likelihood ratio test to find a critical region C of (approximate) size 0.005 (i.e. the probability of a type 1 error is 0.005).
(b) Show that the likelihood ratio test gives you get the same critical region C from part (a) if the alternative hypothesis is H1 : p = p1 where p1 is any number bigger than 1/ 4.
(c) What is the type 2 error associated with the region C when the alternative hypothesis is p = 1/3?
(d) Briefly explain why any other critical region D with size ≤ .005 must make a larger type 2 error than C when the alternative hypothesis is p = 1/3.
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