Page 139 - Applied Statistics And Probability For Engineers

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4-6 NORMAL DISTRIBUTION 117

Mean and Variance of the Normal Distribution (CD Only)

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