Page 110 - Applied Statistics And Probability For Engineers

PQ220 6234F.Ch 03 13/04/2002 03:19 PM Page 88

88 CHAPTER 3 DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

the probability that more than 45 of the sampled customers have purchased from the corpora-
tion in the last three months?
The sampling is without replacement. However, because the sample size of 50 is small
relative to the number of customer accounts, 1000, the probability of selecting a customer who
has purchased from the corporation in the last three months remains approximately constant
as the customers are chosen.
For example, let A denote the event that the first customer selected has purchased
from the corporation in the last three months, and let B denote the event that the second
customer selected has purchased from the corporation in the last three months. Then,
P1A2 700 1000 0.7 and P1B ƒ A2 699 999 0.6997 . That is, the trials are approxi-
mately independent.
Let X denote the number of customers in the sample who have purchased from the cor-
poration in the last three months. Then, X is a hypergeometric random variable with N
1000, n 50, and K 700. Consequently, p K N 0.7 . The requested probability is
P1X 452 . Because the sample size is small relative to the batch size, the distribution of X
can be approximated as binomial with n 50 and p 0.7. Using the binomial approximation
to the distribution of X results in

50 50
x
P1X 452 a a b 0.7 11 0.72 50 x 0.00017
x 46 x
The probability from the hypergeometric distribution is 0.000166, but this requires computer
software. The result agrees well with the binomial approximation.

EXERCISES FOR SECTION 3-8
3-86. Suppose X has a hypergeometric distribution with (d) What is the mean number of unacceptable washers in the
N 100, n 4, and K 20. Determine the following: sample?
(a) P1X 12 (b) P1X 62 3-91. A company employs 800 men under the age of 55.
(c) P1X 42 (d) Determine the mean and variance of X. Suppose that 30% carry a marker on the male chromosome
3-87. Suppose X has a hypergeometric distribution with that indicates an increased risk for high blood pressure.
N 20, n 4, and K 4. Determine the following: (a) If 10 men in the company are tested for the marker in this
(a) P1X 12 (b) P1X 42 chromosome, what is the probability that exactly 1 man
(c) P1X 22 (d) Determine the mean and variance of X. has the marker?
3-88. Suppose X has a hypergeometric distribution with (b) If 10 men in the company are tested for the marker in this
N 10, n 3, and K 4. Sketch the probability mass func- chromosome, what is the probability that more than 1 has
tion of X. the marker?
3-89. Determine the cumulative distribution function for X 3-92. Printed circuit cards are placed in a functional test
in Exercise 3-88. after being populated with semiconductor chips. A lot contains
3-90. A lot of 75 washers contains 5 in which the variability 140 cards, and 20 are selected without replacement for func-
tional testing.
in thickness around the circumference of the washer is unac-
ceptable. A sample of 10 washers is selected at random, (a) If 20 cards are defective, what is the probability that at
without replacement. least 1 defective card is in the sample?
(a) What is the probability that none of the unacceptable (b) If 5 cards are defective, what is the probability that at least
washers is in the sample? 1 defective card appears in the sample?
(b) What is the probability that at least one unacceptable 3-93. Magnetic tape is slit into half-inch widths that are
washer is in the sample? wound into cartridges. A slitter assembly contains 48 blades.
(c) What is the probability that exactly one unacceptable Five blades are selected at random and evaluated each day for
washer is in the sample?