Week_11_Practice_Problem_Solutionss

Week_11_Practice_Problem_Solutionss - Week 11 Practice...

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Week 11 Practice Problems 1. A sample of 200 people are each asked to produce a “random digit” between 1 and 5. The data are value 1 2 3 4 5 frequency 35 42 48 37 38 Does a uniform model with equal probability for each digit fit these data? Step 1: Suppose the uniform model is appropriate Step 2: The multinomial likelihood is 35 42 48 37 38 12345 1 2 3 4 5 (, , , , ) , 1 Lc θθθθθ θθθθθ θθθθθ =+ + + + = With no restrictions, we have ˆˆ ˆ 35 / 200, 42 / 200, 48 / 200, 37 / 200, 38 / 200 θθ θ ===== Assuming the uniform model we have 0.2, 1,. ..,5 j j = = Step 3: The (likelihood ratio) discrepancy measure is 1234 ˆˆˆˆˆ 2ln[ (0.2,0.2,0.2,0.2,0.2) / ( , , , , )] ˆˆˆˆ 2[35ln(0.2 / ) 42ln(0.2 / ) 48ln(0.2 / ) 37ln(0.2 / ) 38ln(0.2 / )] 2.59 dL L 5 ˆ θθθ =− + + + + = Step 4: The p-value is 2 4 ( ) ( 2.59) 0.10 PD d P χ ≥≈ > Step 5: There is no evidence that j is different from 0.2 There is no evidence against the fit of the uniform model. 2. In an ESP experiment, four cards labeled A,B,C,D are placed face down on a table and a subject is asked to peer through the cards in his or her mind and identify which card is which. a) Show that if a subject guesses which card is which, then the probability function for the correct number of matches Y is y 0 1 2 4 ( PY y = ) 9/24 8/24 6/24 1/24
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(4) 11 (4 ) 42 4 4 2 6 (2 ) 4 4 211 1 8 (1 ) 4 1689 (0 ) 1 24 24 24 24 PY == = ⎛⎞ ×× ⎜⎟ ⎝⎠ = ××× = =
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Week_11_Practice_Problem_Solutionss - Week 11 Practice...

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