Introduction to Business Statistics I (QM - 120)
Homework # 5
Problem # 1
In a group of 20 athletes, 6 have used performance-enhancing drugs that are
illegal. Suppose that two athletes are randomly selected from this group. Let x denote
the number of athletes in this sample who have used these illegal drugs. Write the
probability distribution of x. You may draw a tree diagram and use it to write the
probability distribution. Find the mean and the standard deviation of x.
Either you draw a tree diagram with two stages each results two outcomes.
Remember that the probability of the second stage depends on the first one.
2
2
Use E(x) = Σxp(x) and V(x) = Σx p(x) – E(x) for the mean and standard
deviation.
Or, use the hypergeometric distribution to solve this problem where x can
take 0, 1, and 2 with the following parameters:
N = 20, r = 6, and n = 2
You can use the hypergeometric formulas for the mean and variance.
Problem # 2
Let x denote the number of suits owned by a randomly selected CEO of a
corporation. The following table lists the frequency distribution for 1000 CEOs.
x
f
4
70
5
180
6
240
7
210
8
170
9
90
10
40
a. Construct a probability distribution table for the number of suits owned by a
CEO. Graph the probability distribution.
Divide the frequency of each x by 1000 to get each probability
b. Find the following probabilities
i. P(x = 5) ii. P(x > 6) iii. P(4 ≤ x ≤ 7)
c. Find the mean and the standard deviation of x.
Use E(x) = Σxp(x) and V(x) = Σx2p(x) – E(x)2
iv.
P(x ≤ 6)
Problem # 3
An Auto Insurance company insures a certain make of automobile worth
$15,000. In their experience, the following losses occur with the following
probabilities
Loss
Probability
$0
0.80
$1,000
0.10
$5,000
0.05
$10,000
0.03
$15,000
0.02
What premium C should the insurance company charge if it wants the expected gain
to equal $200?
E(x) = [c] x 0.8 + [c – 1000] x 0.1 + … + [c – 15000] x .02 = 200.
Problem # 4
The taste test for PTC (phenylthiocarbamide) is a favorite exercise for every
human genetics class. It has been established that a single gene determines the
characteristics, and that 70% of Americans are “tasters,” while 30% are “non-tasters.”
Suppose that 20 Americans are randomly selected and are tested for PTC.
The binomial conditions are satisfied here. So you may use the binomial
distribution to find the answers with n = 20 and p = .7
a. What is the probability that 17 or more are “tasters”?
b. What is the probability that 15 or fewer are “tasters”?
Problem # 5
The increased number of small commuter airplanes in major airports in the USA
has heightened concerns over air safety. An eastern airport has recorded a monthly
average of five near-misses on landings and takeoffs in the past 5 years.
You can answer this problem using the Poisson distribution. You have to
change the rate of occurrences in one of the questions so both periods will
have the same length.
a. Find the probability that during a given month there are no near-misses on
landings and takeoffs at the airport.
b. Find the probability that during a given month there are five near-misses.
c. Find the probability that there are at least five near-misses.
d. Find the probability that during a three months period there are at most 10
near-misses.
Problem # 6
A candy dish contains five blue, two red, and one green candies. A child
reaches up and selects four candies without looking.
a. What is the probability that there are two blue candies in the selections?
b. What is the probability that the candies all blue?
c. What is the probability that at least one blue candy is in the selection?
d. What is the probability that the green candy is in the selection?
Use the hypergeometric distribution to solve this problem where x can take
1, 2, 3, 4 for the blue balls or 0, 1, 2 for the red balls or 0, 1 for the green
ball with the following parameters:
N = 8, n = 2 and r is the number of successes in the population
depending the question.
Problem # 7
An airline company receives an average of 9.7 complains per day from its
passengers. Find the probability that on a certain day this airline will receive at least 3
complaints, exactly 7 complaints, and no more than 7 complaints.
We have a typical case of a Poisson distribution.
Problem # 8
In a list of 15 households, 9 own homes and 6 do not own homes. Four
households are randomly selected. Find the probability that the number of households
in these 4 who own homes is
a.
exactly 3
b. at most 1
c. less than 2
Use the hypergeometric distribution to solve this problem.
Problem # 9
The QMIS department at Kuwait University has 32 faculty members. Of them,
15 are in favor of the proposition that all CBA students must take a course in
statistical computing and 17 are against this proposition. The department has agreed
to form a committee of 10 faculty members to survey the university administrators,
CBA students, and CEO of major Kuwaiti companies on this issue. If 10 faculty
members are randomly selected from 32, find the probability that the number of
faculty members in this sample who are in favor of the proposition is
a.
exactly 2
b. at most 4
c. at least 5
Again, it is a hypergeometric distribution.
Problem # 10
An average of 8.2 crimes is reported per day to police in a small city. Find the
probability that exactly 3 crimes will be reported to police on a certain day in this city,
at most 2 crimes will be reported to police in the period from 8:00 AM to 02:00 PM.
Use the Poisson distribution to answer this problem. You need to change
the rate of occurrences in one of the questions so both periods will have the
same length.