2. This exercise is to provide more details to the arguments leading to V (r) = (N), nN where (1) n r = -1 and s² = (yi – rx;)², Σ - (2) i=1 see page 8 of Lecture 8. - Let zi = yi — Rx; and consider a simple random sample from a population 21,..., Zn Z1, ZN (3) (4) Let μz be the population mean, σ the population variance, z the sample με mean and s² the sample variance. (a) Show that μz = 0 and express s½ in terms of the quantities in (3). Express in terms by x, y and R. (b) Using the formulas in Lecture 3, find an expression for Ŵ(z) in terms of s, n and N, that is unbiased for V(z). (c) Expand - - με r − R = 1 (r − R) + (1 − ) (r − R). - - (5) με Show that (r - R) = (6) με με It can be argued that the second term in (5) is small compared to the first term. This is because ñ is a good estimator of μx and so is close to μx when n is large. Since R is a constant, it follows that - V (r) = V (r − R) ≈ V(2), (7) where the approximation is due to ignoring the second term in (5).

Give one method to increase statistical power and Which test statistic should you use if you are comparing two variances?

Question 1 iid Consider two samples 1, 2, fa with E(X) = µ and finite V(X) iid 91, 92,, yn fy with E(Y) = y and finite V (Y) = 2. Assuming that the two samples are independent, show that -y is consistent estimator for Hr - Hy. ... = o2, and

Let s be the sample variance of a random sample of size n1 fron a Normal(41,o²) population and s, be the sample variance of a random sample of size 12 from a Nornal(µ2,0²) population and the two samples are independent. Note that the two samples have the same population variance o² but µi # j12 and n1 # 12. Define the pooled variance estimator as (n1 – 1)sỉ + (n12 – 1)s3 T1 + n2 - 2 -- Is s? unbiased for o2 ? Explain with precisc computation. Find the distribution of ºP. Give the reasoning of your answer. Find the variance of s.