Ex7soln - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT 1306 INTRODUCTORY STATISTICS Sem 1 2008 / 2009 EXAMPLE CLASS 7 ANSWERS 1. a) Let Y = the geometric shape is correctly identified by the psychic. Y Bernoulli (p=0.1) For the Bernoulli distribution “Yes” = “1” = “Success” and “No” = “0” = “Failure” and p = probablity of “success” The probability that the psychic guesses correctly is p=0.1. b) Let X = # of geometric shapes correctly identified by the psychic. X Bin (n=10, p=0.1) where n = # of trials and p = probablity of “success” Note: 10 3 2 1 ......... Y Y Y Y X + + + + = A Binomial distribution is the sum of n independent Bernoulli trials. For the Binomial distribution: E(X) = n p and Var(X) = n p (1-p) = n p q Here : E(X) = 10 x 0.1 = 1 and Var(X) = 10 x 0.1 x 0.9 = 0.9. 1
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c) Assuming that the “psychic” is guessing and NOT really reading minds, then P[experimenter will come to a false conclusion that the psychic can read minds] = P[X 8] = P[X=8] + P[X=9] + P[X=10] = 0.000 000 364 5+ 0.000 000 009 + 0.000 000 000 1 = 0.000 000 3731 which is a VERY small chance for this event to occur!!!!! The experimenter is very UNLIKELY to make a false conclusion! Use the binomial formula to find the probabilities: P[X=x] = ( ) x n x n x p p ) 1 ( for x = 0 , 1 , 2 ,……, n P[X=8] = ( ) 8 10 8 10 8 ) 1 . 0 1 ( ) 1 . 0 ( = = 0.000 000 364 5 2 8 ) 9 . 0 ( ) 1 . 0 ( 45 x x P[X=9] = ( ) 9 10 9 10 9 ) 1 . 0 1 ( ) 1 . 0 ( = = 0.000 000 009 1 9 ) 9 . 0 ( ) 1 . 0 ( 10 x x P[X=10] = ( ) 10 10 10 10 10 ) 1 . 0 1 ( ) 1 . 0 ( = = 0.000 000 000 1 0 10 ) 9 . 0 ( ) 1 . 0 ( 1 x x 2
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2. a) Let X=the number of correctly answered questions, in n=10 trials. Questions are answered independently What about the value of p? The wording of this question suggests that the probabilities of answering each question correctly are NOT the same, presumably being smaller for hard questions than for easier questions. Therefore, X does NOT have a binomial distribution.
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Ex7soln - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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