Chapter 6 - Operations Management - Statistics

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Chapter 6 - Continuous Probability Distributions Solutions: 1. a. 3 2 1 .50 1.0 1.5 2.0 f ( x ) x b. P ( x = 1.25) = 0. The probability of any single point is zero since the area under the curve above any single point is zero. c. P (1.0 x 1.25) = 2(.25) = .50 d. P (1.20 < x < 1.5) = 2(.30) = .60 2. a. .15 .10 .05 10 20 30 40 f ( x ) x 0 b. P ( x < 15) = .10(5) = .50 c. P (12 x 18) = .10(6) = .60 d. 10 20 ( ) 15 2 E x + = = e. 2 (20 10) Var( ) 8.33 12 x - = = 3. a. 6 - 1
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Chapter 6 3 / 20 1 / 10 1 / 20 110 120 130 140 f ( x ) x Minutes b. P ( x 130) = (1/20) (130 - 120) = 0.50 c. P ( x > 135) = (1/20) (140 - 135) = 0.25 d. 120 140 ( ) 130 2 E x + = = minutes 4. a. 1.5 1.0 .5 1 2 3 f ( x ) x 0 b. P (.25 < x < .75) = 1 (.50) = .50 c. P ( x .30) = 1 (.30) = .30 d. P ( x > .60) = 1 (.40) = .40 6. a. P ( x 25) = 1 8 (26 – 25) = .125 b. P (21 x 25) = 1 8 (25 – 21) = .50 c. This occurs when programming is 20 minutes or less P ( x 20) = 1 8 (20 – 18) = .25 8. 6 - 2
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Continuous Probability Distributions 10. a. P ( z 1.5) = .9332 b. P ( z 1.0) = .8413 c. P (1 z 1.5) = P ( z 1.5) - P ( z < 1) = .9932 - .8413 = .0919 d. P (0 < z < 2.5) = P ( z < 2.5) - P ( z 0) = .9938 - .5000 = .4938 12. a. P (0 z .83) = .7967 - .5000 = .2967 b. P (-1.57 z 0) = .5000 - .0582 = .4418 c. P ( z > .44) = 1 - .6700 = .3300 d. P ( z -.23) = 1 - .4090 = .5910 e. P ( z < 1.20) = .8849 f. P ( z -.71) = .2389 14. a. The z value corresponding to a cumulative probability of .9750 is z = 1.96. b. The z value here also corresponds to a cumulative probability of .9750: z = 1.96. c. The z value corresponding to a cumulative probability of .7291 is z = .61. d. Area to the left of z is 1 - .1314 = .8686. So z = 1.12. e. The z value corresponding to a cumulative probability of .6700 is z = .44. f. The area to the left of z is .6700. So z = .44. 6 - 3 100 = 10 σ 70 80 90 110 120 130
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Chapter 6 16. a. The area to the left of z is 1 - .0100 = .9900. The z value in the table with a cumulative probability closest to .9900 is z = 2.33. b.
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Chapter 6 - Operations Management - Statistics

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