s-assignment3 - . 4 n = ( 1 . 96 2 . 4 ) 2 = 96 . 04 n 97...

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STAT3502 Solution for assignment # 3 1. a. E ( X ) = x i P ( x i ) = 0 . 2 + 0 . 2 + 0 . 15 = 0 . 55 V ( X ) = x 2 i p ( x i ) - μ 2 = 0 . 2 + 0 . 4 + 0 . 45 - (0 . 55) 2 = 0 . 7475 σ x = 0 . 7475 = 0 . 865 b. Skewed to the right. c. Based on the CLT Shape of distribution of ¯ X is normal (bell shape and symmet- ric). E ( ¯ X ) = μ = 0 . 55 and V ( ¯ X ) = σ 2 n = 0 . 7475 100 = 0 . 007475 d. P ( ¯ X < 0 . 4) = P ( Z < 0 . 4 - 0 . 55 0 . 007475 ) = P ( Z < - 1 . 73) = 0 . 0418 e. P ( X < 0 . 4) = P ( X = 0) = 0 . 65 2. μ = 2 . 8 , n = 25 , ¯ X = 2 . 43 , S = 0 . 2 a. ( ¯ X ± t 0 . 05 , 24 S n ) = (2 . 43 ± 1 . 711( 0 . 2 5 ) the 90% C.I is 2 . 3632 < μ < 2 . 498 ( ¯ X ± t 0 . 005 , 24 S n ) = (2 . 43 ± 2 . 797 0 . 2 5 the 99% C.I is 2 . 318 < μ < 2 . 542 b. 90% C.I is shorter, because it has less confidence. 3. n = 144 , ¯ X = 2386 , S = 400 a. ( ¯ X ± Z α/ 2 S n ) = (2386 ± 2 . 58 400 144 the 99% C.I is 2300 < μ < 2472 b. n = ( 2 . 58 × 400 60 ) 2 = 295 . 84 n 296 4. μ 0 = 10 . 12 , n = 36 , ¯ X = 10 . 98 , S = 3 . 27 a. μ = 10 . 12 vs μ > 10 . 12 so the test statistic is Z = ¯ X - μ S/ n = 10 . 98 - 10 . 12 3 . 27 / 6 = 1 . 578. in 1% level Z 0 . 01 = 2 . 33 and calculated Z < Z α , then “not reject” , in 10% level Z 0 . 1 = 1 . 28 and calculated Z > Z α , then “reject” 5. 95% C.I is 5 . 6 < μ < 6 . 4 a. Point estimate is ¯ X = (6 . 4 + 5 . 6) / 2 = 6 b. σ = 2 , 2 Z α/ 2 σ n = 6 . 4 - 5 . 6 = 0 . 8 , 1 . 96 2 n = 0
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Unformatted text preview: . 4 n = ( 1 . 96 2 . 4 ) 2 = 96 . 04 n 97 c. No assumption, based on CLT. 6. f ( x 1 , , x n , p ) = p x i (1-p ) (1-x i ) ln f ( x 1 , , x n , p ) = x i (ln p ) + (1-x i )(ln(1-p )) 1 ln f ( x 1 , ,x n ,p ) p = x i p- (1-x i ) 1-p = 0 x i p = n- x i 1-p p = x i n = X 7. a. f X ( x ) = R e-( x + y ) dy = e-x P ( X &gt; 1) = R 1 e-x dx = e-1 = 0 . 37 b. P (0 &lt; X &lt; 1 , Y &lt; 2) = R 1 R 2 e-x e-y dydx = 0 . 5465 c. f Y ( y ) = R e-( x + y ) dx = e-y E ( Y ) = R ye-y dy =-ye-y-e-y ] = 1 d. See above. e. Yes, because f ( x, y ) = f X ( x ) f Y ( y ) , e-( x + y ) = e-x e-y 2...
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s-assignment3 - . 4 n = ( 1 . 96 2 . 4 ) 2 = 96 . 04 n 97...

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