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Figure 4-18 Distribution for Example 4-16.
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0.2485 0.25
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4-6 NORMAL DISTRIBUTION
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Mean and Variance of the Normal Distribution (CD Only) EXERCISES FOR SECTION 4-6
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4-39. Use Appendix Table II to determine the following probabilities for the standard normal random variable Z: (a) P(Z 1.32) (b) P(Z 3.0) (c) P(Z 1.45) (d) P(Z 2.15) (e) P( 2.34 Z 1.76) 4-40. Use Appendix Table II to determine the following probabilities for the standard normal random variable Z: (a) P( 1 Z 1) (b) P( 2 Z 2) (c) P( 3 Z 3) (d) P(Z 3) (e) P(0 Z 1) 4-41. Assume Z has a standard normal distribution. Use Appendix Table II to determine the value for z that solves each of the following: (a) P( Z z) 0.9 (b) P(Z z) 0.5 (c) P( Z z) 0.1 (d) P(Z z) 0.9 (e) P( 1.24 Z z) 0.8 4-42. Assume Z has a standard normal distribution. Use Appendix Table II to determine the value for z that solves each of the following: (a) P( z Z z) 0.95 (b) P( z Z z) 0.99 (c) P( z Z z) 0.68 (d) P( z Z z) 0.9973 4-43. Assume X is normally distributed with a mean of 10 and a standard deviation of 2. Determine the following: (a) P(X 13) (b) P(X 9) (c) P(6 X 14) (d) P(2 X 4) (e) P( 2 X 8) 4-44. Assume X is normally distributed with a mean of 10 and a standard deviation of 2. Determine the value for x that solves each of the following: (a) P(X x) 0.5 (b) P(X x) 0.95 (c) P(x X 10) 0.2 (d) P( x X 10 x) 0.95 (e) P( x X 10 x) 0.99 4-45. Assume X is normally distributed with a mean of 5 and a standard deviation of 4. Determine the following: (a) P(X 11) (b) P(X 0) (c) P(3 X 7) (d) P( 2 X 9) (e) P(2 X 8) 4-46. Assume X is normally distributed with a mean of 5 and a standard deviation of 4. Determine the value for x that solves each of the following: (a) P(X x) 0.5 (b) P(X x) 0.95 (c) P(x X 9) 0.2 (d) P(3 X x) 0.95 (e) P( x X x) 0.99 4-47. The compressive strength of samples of cement can be modeled by a normal distribution with a mean of 6000 kilograms per square centimeter and a standard deviation of 100 kilograms per square centimeter. (a) What is the probability that a sample s strength is less than 6250 Kg/cm2 (b) What is the probability that a sample s strength is between 5800 and 5900 Kg/cm2 (c) What strength is exceeded by 95% of the samples 4-48. The tensile strength of paper is modeled by a normal distribution with a mean of 35 pounds per square inch and a standard deviation of 2 pounds per square inch. (a) What is the probability that the strength of a sample is less than 40 lb/in2 (b) If the specifications require the tensile strength to exceed 30 lb/in2, what proportion of the samples is scrapped 4-49. The line width of for semiconductor manufacturing is assumed to be normally distributed with a mean of 0.5 micrometer and a standard deviation of 0.05 micrometer. (a) What is the probability that a line width is greater than 0.62 micrometer (b) What is the probability that a line width is between 0.47 and 0.63 micrometer (c) The line width of 90% of samples is below what value 4-50. The ll volume of an automated lling machine used for lling cans of carbonated beverage is normally distributed with a mean of 12.4 uid ounces and a standard deviation of 0.1 uid ounce. (a) What is the probability a ll volume is less than 12 uid ounces (b) If all cans less than 12.1 or greater than 12.6 ounces are scrapped, what proportion of cans is scrapped (c) Determine speci cations that are symmetric about the mean that include 99% of all cans. 4-51. The time it takes a cell to divide (called mitosis) is normally distributed with an average time of one hour and a standard deviation of 5 minutes. (a) What is the probability that a cell divides in less than 45 minutes (b) What is the probability that it takes a cell more than 65 minutes to divide (c) What is the time that it takes approximately 99% of all cells to complete mitosis 4-52. In the previous exercise, suppose that the mean of the lling operation can be adjusted easily, but the standard deviation remains at 0.1 ounce. (a) At what value should the mean be set so that 99.9% of all cans exceed 12 ounces (b) At what value should the mean be set so that 99.9% of all cans exceed 12 ounces if the standard deviation can be reduced to 0.05 uid ounce
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