Exercise 20. Let X₁,..., Xn~ N(μ, o2) be independent, where u and o are unknown. In lectures we only discussed symmetric confidence intervals. Following the arguments we used in lectures, find an unbounded interval of the form [u, ∞) so that the mean μ in satisfies P(μ € [u, ∞)) = 95%.

Derive that the lower bound of one-sided confidence interval is L = Ô – CaSE[j].

In random, independent samples of 225 adults and 200 teenagers who watched a certain television show, 112 adults and 138 teens indicated that they liked the show. Let p1 be the proportion of all adults watching the show who liked it, and let p2 be the proportion of all teens watching the show who liked it. Find a 95% confidence interval for −p1p2. Then find the lower limit and upper limit of the 95% confidence interval. Carry your intermediate computations to at least three decimal places. Round your responses to at least three decimal places. a.) The Lower Limit is: b.) The Upper Limit is:

4. Let X₁,..., Xn N(μ,2). In this question, we will construct confidence interval for ². Let C~ x²(r), and as usual for any a € [0, 1] we denote x²(r) to be P(C>x²(r)) = a. (a) Suppose that μ is known. By considering the distribution of prove that i.i.d. prove that Xi Σ(^-^)". n i=1 [Σï_₁(Xi −µ²)² Σï-₁ (Xi − µ)²] X²/2(n) Xỉa/z(n) is a 100(1-a) % confidence interval for o². (b) Suppose that is unknown. By considering the distribution of 2 Σ(**) (X-X) (-1).5² = [Σ,(Χ. – X) Σ(Χ. – X)2] X²/2(n-1) ¹X₁-a/2 (n − 1) is a 100(1-a) % confidence interval for o². (c) In the same setting as part (b), that is, suppose that is unknown. Construct a 100(1-a)% confidence interval for o.

Let X1, . . . , Xn ∼ iid N(0, τ ). Find a pivot V (X, τ ) based on a complete sufficient statistic, andusing V (X, τ ) construct a 100γ% confidence interval for τ .

In random, independent samples of 200 adults and 300 teenagers who watched a certain television show, 124 adults and 153 teens indicated that they liked the show. Let p, be the proportion of all adults watching the show who liked it, and let p, be the proportion of all teens watching the show who liked it. Find a 90% confidence interval for p,-p,. Then find the lower limit and upper limit of the 90% confidence interval. Carry your intermediate computations to at least three decimal places. Round your responses to at least three decimal places. (If necessary, consult a list of formulas.) Lower limit: Upper limit: O O