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hw5_sol

# hw5_sol - IEOR 165 Engineering Statistics Quality Control...

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IEOR 165: Engineering Statistics, Quality Control and Forecasting, Summer 2009 Homework 5 Solution Chapter 13 Question 1 Control limits are 35 ± 9 / 5 which give LCL=30.98, UCL=39.03. Subgroup 3 falls outside these limits. Question 2 Suppose the mean jumps to 16.2. The probability that the next subgroup falls outside is approx- imately P { X > 14 + 6 / 5 } = P ( Z > 3 - 2 . 2 5 2 ) = 1 - Φ( . 54) = . 2946 . On average, it will take a geometric distributed number of subgroups with mean 1/.2946=3.39 to detect a shift. The result is the same when the mean falls by 2.2. Question 4 (a) X = 357 . 2 / 25 = 14 . 288, ¯ S = 4 . 88 / 25 = . 1952. Control limits are 14 . 288 ± 3 × . 1952 / ( 5 c (5)) which give LCL=14.01, UCL=14.57. (b) The estimate of σ is .1952/ c (5)=.2077. Hence with μ = 14 . 288, σ = . 2077, P { 13 . 85 < X < 14 . 75 } = P 13 . 85 - 14 . 288 . 2077 X - 14 . 288 . 2077 14 . 75 - 14 . 288 . 2077 = Φ(2 . 224) - Φ( - 2 . 109) = . 969 . Question 7 Noting X = 36 . 23, ¯ S = 5 . 43, c (5) = . 94 (a) Control limits for X are 36 . 23 ± 3 × 5 . 43 / ( 5 c (5)) which give LCL( X )=28.48, UCL( X )=43.98 (b) Control limits for S are 5 . 43[1 ± 3 q 1 /c (5) 2 - 1] which give LCL( S )=0, UCL( S )=11.34 (c) Yes (e) Noting X is normal with mean 36.23 and standard deviation 5.43/.94=5.777 P { 25

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hw5_sol - IEOR 165 Engineering Statistics Quality Control...

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