Exam ST Sample Question Section B.4 Sample Question 1 For a health insurance policy losses follow an unknown continuous distribution with unique median M. You are given the following random sample of ten losses: 7 6 8 12 2 3 13 1 14 15 Based on this sample and the sign test you are asked to test the null hypothesis H0: M = 5.5 against the alternate hypothesis Ha: M < 5.5. Calculate the p-value (significance) of the test. A. Less than 0.05 B. At least 0.05, but less than 0.10 C. At least 0.10, but less than 0.15 D. At least 0.15, but less than 0.20 E. At least 0.20 Exam ST Sample Question Section B.4 Sample Question 1 Solution The number of observed values less than 5.5 follows the binomial distribution with parameters 10, and 0.5. There are 3 observed values less than 5.5 and the p-value is which is 0.172. Answer D ∑30�10 � 0.510 Exam ST Sample Question Section B.4 Sample Question 2 You are given the following samples of paid medical claims for two similar hospitals A, and B. Hospital A 54 56 60 50 59 38 76 12 62 63 57 58 65 61 45 39 Hospital B 87 92 You are asked to use the run test to test the hypothesis that the distributions of paid medical claims are identical for these two hospitals. Calculate the probability of observing the number of runs you find for the above samples. A. Less than 0.10 B. At least 0.10, but less than 0.15 C. At least 0.15, but less than 0.20 D. At least 0.20, but less than 0.25 E. At least 0.25 Exam ST Sample Question Section B.4 Sample Question 2 Solution The combined data once sorted is: 12 63 38 65 39 76 45 87 50 92 54 56 57 58 which results in R=8 runs as follow: AABBAAABBAABBBBABB P(R=8) is Answer B 2�73��39� �18 � 10 = 0.134 59 60 61 62 Exam ST Sample Question Section B.5 Sample Question 1 For a large number of health insurance policies the annual numbers of claims follow the Poisson distribution with mean λ, which varies by policy. The prior probability density function of λ is 1 ℎ() = � � 2 − , 2 In Year I, four claims are reported for a policy. >0 Calculate the expected value of the posterior distribution of λ for this policy. A. Less than 2.0 B. At least 2.0, but less than 3.0 C. At least 3.0, but less than 4.0 D. At least 4.0, but less than 5.0 E. At least 5.0 Exam ST Sample Question Section B.5 Sample Question 1 Solution Pr( = 4|) = − 4 /4! Pr( = 4, ) = − 4 1 2 − � � 4! 2 (| = 4) 6 −2 which is the gamma density with parameters 7 and ½, and expected value of 3.5. Answer is C. Exam ST Sample Question Section B.5 Sample Question 2 For a large number of worker’s compensation policies the annual numbers of claims follow the binomial distribution with parameters n=10 and θ. The parameter θ, which varies by policy, follows the prior probability density function ℎ() = 280 3 (1 − )4 , In Year I, four claims are reported for a policy. 0<<1 Calculate the expected value of the posterior distribution of θ for this policy. A. Less than 0.45 B. At least 0.45, but less than 0.50 C. At least 0.50, but less than 0.55 D. At least 0.55 but less than 0.60 E. At least 0.60 Exam ST Sample Question Section B.5 Sample Question 2 Pr( = 4|) = ( Pr( = 4, ) = ( 10! 4 ) (1 − )6 4! 6! 10! 4 ) (1 − )6 (280) 3 (1 − )4 4! 6! (| = 4) 7 (1 − )10 which is the beta density with parameters 8 and 11 and expected value of 8/19. Answer is A.

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