(Q2) Suppose Y₁,..., Y, are mutually independent random variables such that for a given Xi Y~N(a+BXi, o²), i=1,...,n. where a, ẞ and the variance o² are unknown. Unbiased estimators â, 3, 52 of a, 3, 02 are â=Y-BX, ẞ=S, S²=; B= Sxx' 1 n-2 i=1 (X-&-BX;)², where we have denoted Sry=XY(X) (Y). It is known that - T= B-B S/√√STE ~tn-2- Let c>0 be a constant such that P(|T|> c) = 0.05 where T~tn−2. (i) Show that the 95% confidence interval for ẞ is given by S (ii) Suppose we observe the following data: X 70, 74, 72, 68, 58, 54, 82, 64, 80, 61 Y 77, 94, 88, 80, 71, 76, 88, 80, 90, 69 Calculate, ẞ, S² from the data. (iii) Calculate the 95% confidence interval for ẞ.

Assume X and Y are independent random variables with X ~ Binomial(n =100, p = 0.2) and Y ~ Pascal(m =16, p = 2/3) . Find the indicated expected value and variance. a. E(3X + 2Y +1) b. Var(3X + 2Y +1)

8.3

1. Six batteries taken at random from a large lot were subjected to life tests with the following results: 19, 22, 18, 16, 25, and 20 hours. Find the probability that the sample mean of x= 20 hours, as observed here, or one less than 20 hours could arise if the true mean of the lot of batteries is 21.5 hours, assuming a known variance of o = 10 hours, for the individual batteries in the lot. %3D