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hw8sol - STAT401 Homework 8 Solutions(Spring 2009 7.2.4 1 i...

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STAT401: Homework 8 Solutions (Spring 2009) 7.2.4 1. i. The parameter of interest is the probability that a randomly selected credit card customer in the department store which has incurred an interest charge due to an unpaid balance. a. ˆ p = 136 200 =0.68 b. se(ˆ p )= q ˆ p (1 - ˆ p ) n = q 0 . 68(1 - 0 . 68) 200 =0.0330 c. Yes, E p )=p 4. a. Estimators ˆ θ 1 and ˆ θ 4 are unbiased since E( ˆ θ 1 )=E( ˆ θ 4 )= θ . b. We want the unbiased estimator with the smallest variance (or standard error). Since Var( ˆ θ 1 ) > Var( ˆ θ 4 ), I would choose ˆ θ 4 . 5. Since we have pre-specified number of sample and the selection pro- cedure is independent and identical (i.e., probability of success is identical and independent of others), X Bin ( m, p 1 ) , Y Bin ( n, p 2 ) a. ˆ p 1 = X m , ˆ p 2 = Y n b. E ( ˆ p 1 - ˆ p 2 ) = E ( X m - Y n ) = 1 m E ( X ) - 1 n E ( Y ) = 1 m · mp 1 - 1 n · np 2 = p 1 - p 2 c. s.e. ( d p 1 - p 2 ) = q ˆ p 1 (1 - ˆ p 1 ) m + ˆ p 2 (1 - ˆ p 2 ) n d. d p 1 - p 2 = 60 100 - 150 200 = - 3 20 = - 0 . 15 e. s.e. ( d p 1 - p 2 ) = q 0 . 6 × 0 . 4 100 + 0 . 75 × 0 . 25 200 = 0 . 0024 + 0 . 000938 = 0 . 0578 7. E ( Y | X = x ) = E ( ˆ α 1 + ˆ β 1 x ) = E ( ˆ α 1 ) + E ( ˆ β 1 ) x = α 1 + β 1 x

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