Page 148 - Applied statistics and probability for engineers

126 Chapter 4/Continuous Random Variables and Probability Distributions

EXERCISES FOR SECTION 4-6

Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion.

4-63. Use Appendix Table III to determine the following 4-72. The time until recharge for a battery in a laptop com-
probabilities for the standard normal random variable Z: puter under common conditions is normally distributed with a
(
(a) P Z ,1 32. ) (b) P Z ,3 0) mean of 260 minutes and a standard deviation of 50 minutes.
(
.
(
(
(c) P Z .1 45) (d) P Z . − 2 15 ) (a) What is the probability that a battery lasts more than four
.
.
(e) P − ( 2 34. , Z , 1 76 ) hours?
.
4-64. Use Appendix Table III to determine the following (b) What are the quartiles (the 25% and 75% values) of battery life?
probabilities for the standard normal random variable Z: (c) What value of life in minutes is exceeded with 95% probability?
Z
Z
(a) P − ( 1, , ) 1 (b) P − ( 2, , ) 2 4-73. An article in Knee Surgery Sports Traumatol Arthrosc
(
(c) P − ( 3, , (d) P Z ,3) [“Effect of Provider Volume on Resource Utilization for Surgi-
Z
) 3
(
Z
(e) P 0, , 1) cal Procedures” (2005, Vol. 13, pp. 273–279)] showed a mean
4-65. Assume that Z has a standard normal distribution. time of 129 minutes and a standard deviation of 14 minutes for
Use Appendix Table III to determine the value for z that solves anterior cruciate ligament (ACL) reconstruction surgery at high-
each of the following: volume hospitals (with more than 300 such surgeries per year).
(
(
.
.
(a) P Z , z) = 0 9 (b) P Z , z) = 0 5 (a) What is the probability that your ACL surgery at a high-
(
(
.
.
(c) P Z . z) = 0 1 (d) P Z . z) = 0 9 volume hospital requires a time more than two standard
(e) P − ( 1 24. , Z , z) = 0 8 deviations above the mean?
.
4-66. Assume that Z has a standard normal distribution. (b) What is the probability that your ACL surgery at a high-
Use Appendix Table III to determine the value for z that solves volume hospital is completed in less than 100 minutes?
each of the following: (c) The probability of a completed ACL surgery at a high-vol-
(a) P − ( z , , z) = 0 95 (b) P − ( z , , z) = 0 99 ume hospital is equal to 95% at what time?
Z
Z
.
.
(c) P − ( z , Z , z) = 0 68 (d) P − ( z , , z) = 0 9973 (d) If your surgery requires 199 minutes, what do you conclude
Z
.
.
4-67. Assume that X is normally distributed with a mean about the volume of such surgeries at your hospital? Explain.
of 10 and a standard deviation of 2. Determine the following: 4-74. Cholesterol is a fatty substance that is an important part
(
(a) P Z ,13) (b) P Z . 9) of the outer lining (membrane) of cells in the body of animals.
(
(
(
(c) P 6, X , 14) (d) P 2, X , 4) (e) P − ( 2, X , ) 8 Its normal range for an adult is 120–240 mg/dl. The Food and
4-68. Assume that X is normally distributed with a mean Nutrition Institute of the Philippines found that the total cho-
of 10 and a standard deviation of 2. Determine the value for x lesterol level for Filipino adults has a mean of 159.2 mg/dl and
that solves each of the following: 84.1% of adults have a cholesterol level less than 200 mg/dl
(
(
(a) P X . x) = 0 5. (b) P X . x) = 0 95 (http://www.fnri.dost.gov.ph/). Suppose that the total choles-
.
(
.
(c) P x , X ,10) = 0. (d) P − ( x , X −10 , x) = 0 95 terol level is normally distributed.
(e) P − ( x , X −10 , x) = 0 99 (a) Determine the standard deviation of this distribution.
.
4-69. Assume that X is normally distributed with a mean (b) What are the quartiles (the 25% and 75% percentiles) of
of 5 and a standard deviation of 4. Determine the following: this distribution?
(
(
(a) P X ,11) (b) P X . 0) (c) P 3, X , 7) (c) What is the value of the cholesterol level that exceeds 90%
(
(
(d) P − ( 2, X , ) 9 (e) P 2, X , 8) of the population?
4-70. Assume that X is normally distributed with a mean of (d) An adult is at moderate risk if cholesterol level is more
5 and a standard deviation of 4. Determine the value for x that than one but less than two standard deviations above the
solves each of the following: mean. What percentage of the population is at moderate
(
(
.
(a) P X . x) = 0 5. (b) P X . x) = 0 95 risk according to this criterion?
(
(
(c) P x , X , 9) = 0 2 (d) P 3, X , x) = 0 95 (e) An adult whose cholesterol level is more than two standard
.
.
(e) P − ( x , X − , x) = 0 99 deviations above the mean is thought to be at high risk.
5
.
What percentage of the population is at high risk?
4-71. The compressive strength of samples of cement can
(f) An adult whose cholesterol level is less than one standard
be modeled by a normal distribution with a mean of 6000 kilo-
deviations below the mean is thought to be at low risk.
grams per square centimeter and a standard deviation of 100
What percentage of the population is at low risk?
kilograms per square centimeter.
(a) What is the probability that a sample’s strength is less than 4-75. The line width for semiconductor manufacturing is
2
6250 Kg/cm ? assumed to be normally distributed with a mean of 0.5 microm-
(b) What is the probability that a sample’s strength is between eter and a standard deviation of 0.05 micrometer.
2
5800 and 5900 Kg/cm ? (a) What is the probability that a line width is greater than 0.62
(c) What strength is exceeded by 95% of the samples? micrometer?