Page 154 - Applied statistics and probability for engineers

132 Chapter 4/Continuous Random Variables and Probability Distributions

EXERCISES FOR SECTION 4-7

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

4-95. Suppose that X is a binomial random variable with 4-103. Suppose that the number of asbestos particles in
0
n = 200 and p = . . Approximate the following probabilities: a sample of 1 squared centimeter of dust is a Poisson random
4
(a) P X( ≤ 70 ) (b) P(70 < X < 90) (c) P(X = 80) variable with a mean of 1000. What is the probability that 10
4-96. Suppose that X is a Poisson random variable squared centimeters of dust contains more than 10,000 particles?
with λ = 6. 4-104. A high-volume printer produces minor print-quality
(a) Compute the exact probability that X is less than four. errors on a test pattern of 1000 pages of text according to a Pois-
(b) Approximate the probability that X is less than four and son distribution with a mean of 0.4 per page.
compare to the result in part (a). (a) Why are the numbers of errors on each page independent
(c) Approximate the probability that 8 < X < 12. random variables?
4-97. Suppose that X has a Poisson distribution with a (b) What is the mean number of pages with errors (one or more)?
(c) Approximate the probability that more than 350 pages con-
mean of 64. Approximate the following probabilities:
(
(
(
(a) P X > 72) (b) P X < 64) (c) P 60 < X ≤ 68) tain errors (one or more).
4-105. Hits to a high-volume Web site are assumed to follow
4-98. The manufacturing of semiconductor chips pro-
a Poisson distribution with a mean of 10,000 per day. Approxi-
duces 2% defective chips. Assume that the chips are inde-
mate each of the following:
pendent and that a lot contains 1000 chips. Approximate the
(a) Probability of more than 20,000 hits in a day
following probabilities:
(b) Probability of less than 9900 hits in a day
(a) More than 25 chips are defective.
(c) Value such that the probability that the number of hits in a
(b) Between 20 and 30 chips are defective.
day exceeds the value is 0.01
4-99. There were 49.7 million people with some type of (d) Expected number of days in a year (365 days) that exceed
long-lasting condition or disability living in the United States 10,200 hits.
in 2000. This represented 19.3 percent of the majority of civil- (e) Probability that over a year (365 days), each of the more
ians aged ive and over (http://factinder.census.gov). A sample than 15 days has more than 10,200 hits.
of 1000 persons is selected at random. 4-106. An acticle in Biometrics [“Integrative Analysis of
(a) Approximate the probability that more than 200 persons in Transcriptomic and Proteomic Data of Desulfovibrio Vulgaris:
the sample have a disability. A Nonlinear Model to Predict Abundance of Undetected Pro-
(b) Approximate the probability that between 180 and 300 teins” (2009)] reported that protein abundance from an operon
people in the sample have a disability. (a set of biologically related genes) was less dispersed than
4-100. Phoenix water is provided to approximately 1.4 million from randomly selected genes. In the research, 1000 sets of
people who are served through more than 362,000 accounts (http:// genes were randomly constructed, and of these sets, 75% were
phoenix.gov/WATER/wtrfacts.html). All accounts are metered and more disperse than a speciic opteron. If the probability that
billed monthly. The probability that an account has an error in a a random set is more disperse than this opteron is truly 0.5,
month is 0.001, and accounts can be assumed to be independent.
approximate the probability that 750 or more random sets
(a) What are the mean and standard deviation of the number of
exceed the opteron. From this result, what do you conclude
account errors each month?
about the dispersion in the opteron versus random genes?
(b) Approximate the probability of fewer than 350 errors in a month.
4-107. An article in Atmospheric Chemistry and Physics
(c) Approximate a value so that the probability that the number
[“Relationship Between Particulate Matter and Childhood Asthma
of errors exceeds this value is 0.05.
– Basis of a Future Warning System for Central Phoenix,” 2012,
(d) Approximate the probability of more than 400 errors per Vol. 12, pp. 2479-2490] linked air quality to childhood asthma
month in the next two months. Assume that results between incidents. The study region in central Phoenix, Arizona recorded
months are independent. 10,500 asthma incidents in children in a 21-month period. Assume
4-101. An electronic ofice product contains 5000 elec- that the number of asthma incidents follows a Poisson distribution.
tronic components. Assume that the probability that each compo- (a) Approximate the probability of more than 550 asthma inci-
nent operates without failure during the useful life of the product dents in a month.
is 0.999, and assume that the components fail independently. (b) Approximate the probability of 450 to 550 asthma inci-
Approximate the probability that 10 or more of the original 5000 dents in a month.
components fail during the useful life of the product. (c) Approximate the number of asthma incidents exceeded
4-102. A corporate Web site contains errors on 50 of 1000 with probability 5%.
pages. If 100 pages are sampled randomly without replace- (d) If the number of asthma incidents was greater during the
ment, approximate the probability that at least one of the pages winter than the summer, what would this imply about the
in error is in the sample. Poisson distribution assumption?