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- Let Y₂ represent the ith normal population with unknown mean 4, and unknown variance of for i=1,2. Consider independent random samples, Y₁₁, Yi2,,Yin, of size ni, from the ith population with sample mean Y, and sample variance S?=²-1₁-1(Y - Y₁². (a) What is the distribution of Y;? State all the relevant parameters of the distribution. (b) Find a level a test (that is, the rejection region) for testing Ho: ₁ = o versus Ha Hiio when of is unknown and n; is small. (c) In the context of the test in part (b), state the Type I error and give a probability statement for the level of significance, a.Let Y, represent the ith normal population with unknown mean #, and unknown variance of for i=1,2. Consider independent random samples, Ya, Ya, Yin, of size n,, from the ith population with sample mean Y, and sample variance S? =(₁-₁)². j=1 (e) Is V₁ a pivotal function to find a confidence interval for o?? Explain with argument. (f) Find a (1-a) x 100% confidence interval for of (g) For non-zero constants a's, what is the distribution of U₂ = a₁Y₁-0₂Y₂? State all the relevant parameters of the distribution.Calculate value of (t) if: first sample mean = 85, second sample mean = 91, and o x1-x2 = 3.
- Q2) Let X, X X. iid N(H,0), be a random sample from the normal population where u is assumed known. Please derive (a) The maximum likelihood estimator (MLE) for o'. (b)Is the MLE an efficient estimator foro? (c) Is the sample variance an efficient estimator for o ?Every year, the students at a school are given a musical aptitude test that rates them from 0 (no musical aptitude) to 5 (high musical aptitude). This year's results were: Aptitude Score 0 1 2 3 4 5 Frequency 4 1 1 4 4 1 The mean (T) aptitude score: The median aptitude score: (Please show your answer to 1 decimal place.) (Please separate your answers by ',' in bimodal situation. Enter DNE The mode aptitude score: if there is no mode.) Textbook Measures of CenterLet Y, represent the ith normal population with unknown mean 44, and unknown variance of for i 1,2. Consider independent random samples, Ya, Y₁2,,Yin, of size ni, from the ith population with sample mean Y, and sample variance S?=1Σj=1(Y₁j - Y₁². (a) What is the distribution of Y;? State all the relevant parameters of the distribution. (b) Find a level a test (that is, the rejection region) for testing Ho : μi = μio versus Ha: Pipio when of is unknown and n, is small. (c) In the context of the test in part (b), state the Type I error and give a probability statement for the level of significance, a.
- Let Y, represent the ith normal population with unknown mean , and unknown variance of for i=1,2. Consider independent random samples, Y₁₁, Y2Yin, of size n,, from the ith population with sample mean Y, and sample variance S?=-(Yu-Y)² (a) What is the distribution of Y,? State all the relevant parameters of the distribution. (b) Find a level a test (that is, the rejection region) for testing Ho: 4 = 40 versus Ha Pio when of is unknown and n, is small. (e) In the context of the test in part (b), state the Type I error and give a probability statement for the level of significance, a. (d) Define V₁, a function of S7, that has a chi-square distribution.Let X1, ... , Xn be a random sample (i.i.d.) from Geometric(p) distribution with PMF P(X = x) = (1 – p)*p, x = 0, 1, 2, . .. The mean of this distribution is (1 – p)/p. (a) :) Find the MLE of p. (b) ( s) Find the estimator for p using method of moments. (c) Now let's think like a Bayesian. Consider a Beta prior on p, i.e., p~ Beta(a, B). Find the posterior distribution of p. Hint: For Geometric likelihood, the conjugate prior on p is a Beta distribution. (d) ) What is the Bayes estimator of p under squared error loss? Denote it by PB. (e) What happens to på if both a and ß goes to 0?Let Y, represent the ith normal population with unknown mean , and unknown variance of for i=1,2. Consider independent random samples, Ya, Y2. the ith population with sample mean Y, and sample variance S² = Yin, of size n,, from (Y-₁². (a) What is the distribution of Y,? State all the relevant parameters of the distribution. (b) Find a level a test (that is, the rejection region) for testing Ho : 4 = o versus Ha i Pio when of is unknown and n, is small. : (e) In the context of the test in part (b), state the Type I error and give a probability statement for the level of significance, a.
- Q3) Let X1,X2,. X, be a random sample from the truncated exponential distribution with pdf: f(x,0) = Please else (a) Derive the method of moment estimator of 0; (b) Derive the MLE of 0. (c) Are the MOME and the MLE unbiased estimators for 0? (d) Compare the MSE of the MOME and the MLE for 0. Which one is a better estimator for 0Let I be an indicator variable with p(I) = {p , I=1 } {1-p , I=0}. Show that the variance of I is p(1-p)Let X1,X2,.,X, be a random sample from f(x;8) = 0e-8xIq0,c0) (x). Find a 100y or (1 - a)100 percent confidence interval for the mean of the population.