Chapter 3 Answers - Chapter 3 Answers 1 The central limit...

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Chapter 3 Answers 1. The central limit theorem suggests that when the sample size ( n ) is large, the distribution of the sample average ( Y ) is approximately 2 , Y Y N μ σ with 2 2 . Y n Y σ σ = Given a population 100, Y μ = 2 43 0, Y σ = . we have (a) 100, n = 2 2 43 100 0 43, Y n Y σ σ = = = . and 100 101 100 Pr( 101) Pr (1.525) 0 9364 0 43 0 43 Y Y - - < = < ≈ Φ = . . . . (b) 64, n = 2 2 43 64 64 0 6719, Y Y σ σ = = = . and 101 100 100 103 100 Pr(101 103) Pr 0 6719 0 6719 0 6719 (3 6599) (1 2200) 0 9999 0 8888 0 1111 Y Y - - - < < = < < . . . ≈ Φ . - Φ . = . - . = . . (c) 165, n = 2 2 43 165 0 2606, Y n Y σ σ = = = . and 100 98 100 Pr( 98) 1 Pr( 98) 1 Pr 0 2606 0 2606 1 ( 3 9178) (3 9178) 1 0000 (rounded to four decimal places) Y Y Y - - = - = - . . - Φ - . = Φ . = . . 2. Each random draw i Y from the Bernoulli distribution takes a value of either zero or one with probability Pr ( 1) i Y p = = and Pr ( 0) 1 . i Y p = = - The random variable i Y has mean ( ) 0 Pr( 0) 1 Pr( 1) , i E Y Y Y p = × = + × = = and variance 2 2 2 2 2 var( ) [( ) ] (0 ) Pr( 0) (1 ) Pr( 1) (1 ) (1 ) (1 ) i i Y Y E Y p Y p Y i i p p p p p p μ = - = - ä = + - = = - + - = - .
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(a)The fraction of successes is 1 ( 1) (success) n i i i Y # Y # p Y n n n = = = = = = . (b) 1 1 1 1 1 ( ) ( ) n n n i i i i i Y E p E E Y p p n n n = = = ° = = = = . (c) 1 2 2 1 1 1 1 (1 ) var( ) var var( ) (1 ) n n n i i i i i Y p p p Y p p n n n n = = = ° - = = = - = . The second equality uses the fact that 1 Y , , Y n are i.i.d. draws and cov( , ) 0, i j Y Y = for . i j 3. Denote each voter’s preference by . Y 1 Y = if the voter prefers the incumbent and 0 Y = if the voter prefers the challenger. Y is a Bernoulli random variable with probability Pr ( 1) Y p = = and Pr ( 0) 1 . Y p = = - From the solution to Exercise 3.2, Y has mean p and variance (1 ). p p - (a) 215 400 0 5375. p = = . (b) ˆ ˆ (1 ) 0.5375 (1 0.5375) 4 400 ˆ var( ) 6 2148 10 . p p n p - - - = = = . The standard error is SE 1 2 ( ) (var( )) 0 0249. p p = = . (c)The computed t -statistic is 0 0 5375 0 5 1 506 SE( ) 0 0249 p act p t p μ , - . - .
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