STAT 230 Final, Fall 2005

STAT 230 Final Exam Feb 1, 2006 Time = 1 hour 1. Let X = (x1, · · · , xm)′ be a random sample from the pdf f(x; σ) = 1 σ √ 2π exp[− 1 2σ2(x − 10)2] If �20 i=1 Xi = 180 and �20 i=1 X2 i = 2000, find a 90% confidence interval for σ. 2. Let X be a Bernoulli random variable with a pdf f(x; θ) = θx(1 − θ)1−x if x = 0 or 1 and 0 < θ < 1 We would like to test the hypothesis H0 : θ = 1 2 vs Ha : θ = 2 3 It is agreed to perform two observations X1 and X2 . If both X1 = 1 and X2 = 1 then we reject H0; otherwise we don’t reject H0. Find α = P(Type I error) and β = P(Type II error). 3. If X has the pdf f(x; µ) = 1 10 √ 2π exp[−1 2(x − µ 10 )2], How large should n be chosen so that when testing H0 : µ = 100 vs Ha : µ = 110 the values of α = 0.05 and β = 0.10? 4. A box is known to contain either 3 red and 5 black balls or 5 red and 3 black balls. Three balls are drawn randomly and without replacement. If three red balls are obtained, the decision will be 5 red and 3 black; otherwise, the decision will be 3 red and 5 black balls. Calculate the values of α and β. 1