2. Let X1,..., X, are iid Poisson() random variables. (a) Find a level a large sample...

Transcribed Image Text:

2. Let X1,..., X, are iid Poisson() random variables. (a) Find a level a large sample likelihood ratio test for testing Ho : \ . = o versus Ha : \ > i=1 (b) Suppose that in a survey of 35 people from the university campus, people were asked how many times they have used ChatGpt in the last week for their school's work. If we denote the response from the ith subject as Xi, then we found that 31 X; = 109. Let A be the average number of times ChatGPT was used in the last week by any randomly chosen person from the university campus. Apply the above result of 2a) to test i) Ho: \ 2.5 versus Ha > 2.5 at the 5% level, ii) by inverting the test construct a 95% CI for X, iii) calculate the power of the test when = 3, v) what would be the required sample size if we want 90% power when = 4. = : (c) Find a level a large sample likelihood ratio test for testing Ho : = 0 versus H : \ < 10. Calculate the power of the test when o 2, the true parameter = 1, n = 40 and a = = : 0.01. Calculate the power of the test when 0 = 2, the true parameter = 1, n = 40 and a = : 0.05. (d) Find a level a large sample likelihood ratio test for testing Ho : = 0 versus H : No. Calculate the power of the test when 0 = 2, the true parameter \ = 1, n = 40 and a = 0.01. Suppose that for a dataset of n = 40 observations, /11 Xi = 95. Calculate the p-value of the test for testing H : \ = 2 versus Ha 2 at a = 0.01. = 2. Let X1,..., X, are iid Poisson() random variables. (a) Find a level a large sample likelihood ratio test for testing Ho : \ . = o versus Ha : \ > i=1 (b) Suppose that in a survey of 35 people from the university campus, people were asked how many times they have used ChatGpt in the last week for their school's work. If we denote the response from the ith subject as Xi, then we found that 31 X; = 109. Let A be the average number of times ChatGPT was used in the last week by any randomly chosen person from the university campus. Apply the above result of 2a) to test i) Ho: \ 2.5 versus Ha > 2.5 at the 5% level, ii) by inverting the test construct a 95% CI for X, iii) calculate the power of the test when = 3, v) what would be the required sample size if we want 90% power when = 4. = : (c) Find a level a large sample likelihood ratio test for testing Ho : = 0 versus H : \ < 10. Calculate the power of the test when o 2, the true parameter = 1, n = 40 and a = = : 0.01. Calculate the power of the test when 0 = 2, the true parameter = 1, n = 40 and a = : 0.05. (d) Find a level a large sample likelihood ratio test for testing Ho : = 0 versus H : No. Calculate the power of the test when 0 = 2, the true parameter \ = 1, n = 40 and a = 0.01. Suppose that for a dataset of n = 40 observations, /11 Xi = 95. Calculate the p-value of the test for testing H : \ = 2 versus Ha 2 at a = 0.01. =