1. Let Y1,..., Y, be a random sample from a normal distribution with mean u and variance 4. (a) Give the (sampling) distribution of the sample mean (Y). (b) Suppose that for testing Ho : µ < 10 against Ha: µ > 10, a test rejects Ho if and only if Ý > k, where k is a given constant. Let P(reject Ho) denote the probability that the test rejects Ho when the true mean is µ. (i) Express P(reject Ho) in terms of n, k, µ and the standard normal cdf .

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?=₁1(Yu - Y.². (g) For non-zero constants a,'s, what is the distribution of U₂ = a₁Y₁-a₂Y₂? State all the relevant parameters of the distribution. (h) Find the standard error of U₂ in part (g), assuming that of = o2 = 0². (i) Discuss how the distribution of Y₁ - ₂ can be used to test the equality of the two population means, #, and p2, when of=o=o² is known. (j) Define appropriate rejection regions, in terms of Y₁ - ₂, for testing Ho: #₁ = ₂ against a two-sided alternative hypothesis at the a level of significance.

Let Y, represent the ith normal population with unknown mean 4, and unknown variance of for i=1,2. Consider independent random samples, Ya, Y₁2, Yin, of size n,, from the ith population with sample mean Y, and sample variance S?=1 Σ₁-1 (₁-₁)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: #₁ = μio versus Ha Hi Ho 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.

3 For an SRS {4₁};=, from the normal distribution. ==// (Y-M) 2 fy (y;u, 0²) - 1 е värvoze Show that the sample mean I is a consistent estimator,

Let Y1, Y2,..., Y; be a random sample of size 5 from a normal population with mean 0 and variance 1 and let Y = (1/5) Y;. Let Y, be another independent observation from the same population. What is the distribution of 7.37 a W = E Y? Why? b U = E- (Y; – Y)?? Why? c E- (Y; – Y)² + Y?? Why? i3D1

Suppose X, Y, Z are i.i.d. from a Normal distribution with mean μ and variance 4. You have to compare between three estimators for μ, which are    T1:= (2X + Y)/3 , T2:= (X + Y − Z)/2 , T3:= (Y + 2Z)/2 .   (a) Which among the above are unbiased estimators for μ? (b) Which among these have the smallest variance? (c) Can you propose an estimator for μ which is better than the 3 above? You have to show that your estimator is indeed better.

Let Y, represent the th normal population with unknown mean 4, and unknown variance of for i=1,2. Consider independent random samples, Y₁, Y2. Yin, of size n₁, from the ith population with sample mean Y, and sample variance S² = ₁₁-1(Y₁-₁². j=1 (g) For non-zero constants a's, what is the distribution of U₂ = a₁Y₁-a₂Y₂? State all the relevant parameters of the distribution. (h) Find the standard error of U₂ in part (g), assuming that of = 0² = 0². (i) Discuss how the distribution of Y₁ - ₂ can be used to test the equality of the two population means, #₁ and μ2, when o² = 0 = 0² is known. (j) Define appropriate rejection regions, in terms of Y₁ - Y2, for testing Ho: #₁ = 2 against a two-sided alternative hypothesis at the a level of significance.

Let Y, represent the ith normal population with unknown mean 4, 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? = E(YY)². 7-1 (h) Find the standard error of U₂ in part (g), assuming that of = 0} = 0². (i) Discuss how the distribution of Y₁ - Y₂ can be used to test the equality of the two population means, ₁ and 2, when of = 0 = 0² is known. (j) Define appropriate rejection regions, in terms of Y₁ -Y2, for testing Ho: ₁ = H₂ against a two-sided alternative hypothesis at the a level of significance.

Let X1, X2,...Xn be a random sample from a Poisson distribution with variance 0. Find an unbiased estimator for Pe(X=0).