Statistics Exam: Spring 2009, Statistics 131C, Sample Final, Exams of Mathematical Statistics

Download Statistics Exam: Spring 2009, Statistics 131C, Sample Final and more Exams Mathematical Statistics in PDF only on Docsity! Spring 2009 Statistics 131C Sample Final (Show all the relevant works) 1. Suppose that X1, . . . , Xn form a random sample from a Poisson distribution with mean λ. Let λ1 > λ0 > 0. Suppose that you want to test the following hypotheses: H0 : λ = λ0 against H1 : λ = λ1. (a) Show that the value of α(δ) + β(δ) is minimized by a test procedure which rejects H0 when Xn > c. (b) Find the value of c. (c) Based on your answer in part (a), find (with justification) a nonrandomized UMP level α0 test for H0 : λ ≤ λ0 against H1 : λ > λ0. Can this test be found for all α0 ∈ (0, 1) ? 2. Consider two different normal distributions for which both the means µ1 and µ2 and the variances σ21 and σ 2 2 are unknown. Suppose that a random sample consisting of 16 observations from the first normal population yields ∑16 i=1 Xi = 84 and ∑16 i=1 X 2 i = 563. An independent random sample consisting of 10 observations from the second random sample yields ∑10 i=1 Yi = 18 and ∑10 i=1 Y 2 i = 72. (a) What are the MLE’s of σ21 and σ 2 2 ? (b) Test the following hypotheses at α0 = 0.05: H0 : σ21 ≤ σ22 against H1 : σ21 > σ22. 3. Suppose that 300 persons are selected at random from a large population, and each person in the sample is classified according to blood type, O, A, B or AB; and also according to Rh factor, positive or negative. The data are given below: O A B AB Rh positive 82 89 54 19 Rh negative 13 27 7 9 At 0.05 level of significance, test the hypothesis that the two classifications of blood types are independent. State clearly the model, hypothesis, test statistic and final conclusion. 4. Data on different varieties of seafood in a certain market for the years 1970 and 1980 are reported below. 1