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Uppsala University Department for Information Science Statistics B1 2012-11-14 WRITTEN EXAM FOR THE COURSE, PROBABILITY THEORY AND STATISTICAL INFERENCE, B1 (7.5 ECTS) Writing time: 0800-1200 Permitted aids: Formulas for the course Probability Theory and Statistical Inference Math-Handout, Lars Forsberg Pocket calculator Dictionary (or word-list) Notations in the permitted aids are not allowed. The written examination has 4 problems, for a total of 100 points. If you desire clarification regarding the test, especially the wording of a problem, then please alert an examination proctor. The examination proctors can contact the responsible instructor. After turning in your test, you may keep the test-pages with the question-statements. INSTRUCTIONS: A. Carefully follow the instructions that are listed on the examination-directions page. B. State the assumptions that must be made for the method to be applicable. C. Account for every essential step in your solution. If special concerns are raised in the problem statement, then your solution must carefully address those concerns. 53/12 Task 1. (20p) Bowl B1 contains 2 white chips, bowl B2 contains 2 red chips, bowl B3 contains 2 white and 2 red chips, and bowl B4 contains 3 white and 1 red chips. The probabilities of selecting bowl B1, B2, B3 or B4 are 1/2, 1/4, 1/8, and 1/8, respectively. A bowl is randomly selected, and a chip is then drawn at random. a) Find P(W), the probability of drawing a white chip (W denotes white color). b) Find P(B4|W), the conditional probability that bowl B4 had been selected, given that a white chip was drawn. c) Find P(B4|R) the conditional probability that bowl B4 had been selected, given that a red chip was drawn. Task 2 (30) Given three independent random samples where for sample 1: Y1 is bin( n1 = 4, p = 0.2), for sample 2: Y2 is bin( n2 = 6, p = 0.2) and for sample 3: Y3 is bin( n3 = 10, p = .2). a) What is the probability that Y1 is larger than one? b) What is the probability that Y1, Y2 and Y3 all are larger than one? c) What is the probability that the sum Z= Y1 + Y2 + Y3 is larger than three? d) What is the correlation between Y1 and Z? Task 3. (20) Let Y1 ,Y2 ,Yn denote a random sample from the probability density function 2 ( 1) y , 0 y 1, 2 f ( y | ) 2 0, elsewhere a) Find the Maximum likelihood estimator for b) Find the Maximum likelihood estimator for θ. 53/12 2 . Task 4. (30p) Below you find two distributions. The one to the left is the sampling distribution for the sample mean when the null hypothesis is true (µ=25, 𝜎𝑦̅ = 1, n=40). The second one is the sampling distribution for a hypothesized mean of 27. Assume that you reject the null hypothesis if the sample mean >27. a) Specify the hypothesis, and assumptions for the test. b) What is the significance level of the test. c) If the true mean is 27, what is the probability of a type II error. d) If you want the probability of the type II error to be less than 0.20. How many observations are required in order to detect the presented difference (true mean = 27 ). Assume the same significance level as in task b). Distribution Plot Normal; StDev=1 Mean 25 27 0,4 Density 0,3 0,2 0,1 0,0 22 23 24 25 26 27 X 53/12 28 29 30 31