ISyE 6739 — Test 3 Solutions — Summer 2011

831. Suppose X 1 , . . . , X 5 are i.i.d. normal with unknown mean and unknown varianceσ 2 . Further suppose that ¯X = 100 and S 2 = 25. Find a 95% two-sided confidenceinterval for σ 2 .Solution:⎡σ 2 (n − 1)S2∈ ⎣ ,χ 2 α2 ,n−1⎤(n − 1)S2⎦ =χ 2 1− α 2 ,n−1[ ] 10011.14 , 1000.48= [8.98, 208.33]. ♦32. Consider i.i.d. normal observations X 1 , . . . , X 5 with unknown mean µ and unknownvariance σ 2 . What is the expected width of the usual 90% two-sided confidenceinterval for σ 2 ? You can keep your answer in terms of σ.Solution: The confidence interval is of the form⎡⎤σ 2 (n − 1)S2 (n − 1)S2∈ ⎣ , ⎦ .χ 2 α2 ,n−1 χ 2 1− α 2 ,n−1Thus, the length isL =and so the expected length is⎡1L = (n − 1) ⎣ − 1χ 2 1− α 2 ,n−1 χ 2 α⎡⎣(n − 1)S2χ 2 1− α 2 ,n−1 −2 ,n−1 ⎤⎤(n − 1)S2⎦ ,χ 2 α2 ,n−1[⎦ 1· E[S 2 ] = 40.71 − 19.49]σ 2 = 5.21σ 2 . ♦33. Suppose we conduct an experiment to test to see if people can throw farther rightorleft-handed. We get 20 people to do the experiment. Each throws a ball righthandedonce and throws a ball left-handed once, and we measure the distances. Ifwe are interested in determining a confidence interval for the mean difference inleft- and right-handed throws, which type of c.i. would we likely use?(a) z (normal) confidence interval for differences(b) pooled t confidence interval for differences(c) paired t confidence interval for differences(d) χ 2 confidence interval for differences