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25. The mean and the standard deviation of an exponential distribution are equal to each other.
26. If the random variable X is exponentially distributed and the parameter of the distribution = 4, then
P(X 0.25) = 0.3679.
27. Given that z is a standard normal random variable, a negative value of z indicates that the standard
deviation of z is negative.
28. The mean of any normal distribution is always zero.
29. The exponential distribution is suitable to model the length of time that elapses before the first
telephone call is received by a switchboard.
30. The values of zA are the 100(1 – A)th percentiles of a standard normal random variable.
31. In the standard normal distribution, z0.05 = 1.645 means that there is a 5% chance that the standard
normal random variable Z assumes a value above 1.645.
32. The normal approximation to the binomial distribution works best when the number of trials is large,
and when the binomial distribution is symmetrical (like the normal).
SHORT ANSWER
1. A certain brand of flood lamps has a lifetime that is normally distributed with a mean of 3750 hours
and a standard deviation of 300 hours.
a. What proportion of these lamps will last for more than 4000 hours?
b. What lifetime should the manufacturer advertise for these lamps in order that only 2% of the lamps
will burn out before the advertised lifetime?
2. If X is a normal random variable with a mean of 78 and a standard deviation of 5, find the following
probabilities:
a. P(X 87).
b. P(X 91).
c. P(70 X 77).
3. A continuous random variable X has the probability density function f(x) = (4 – x)/8, 0
x
4. Find
the following probabilities:
a. P(X
1).
b. P(X
2).
c. P(1
X
2).
d. P(X = 3).
4. If the random variable X is normally distributed with a mean of 70 and a standard deviation of 10, find
the following values of the distribution of X.
a. First quartile.
b. Third quartile.
5. Researchers studying the effects of a new diet found that the weight loss over a one-month period by
those on the diet was normally distributed with a mean of 7 kg and a standard deviation of 2.5 kg.
a. What proportion of the dieters lost more than 10 kg?
b. What proportion of the dieters gained weight?
c. If a dieter is selected at random, what is the probability that the dieter lost at most 5 kg?
6. Let X be an exponential random variable with = 2.50. Find the following probabilities.
a. P(X
1.5).
b. P(X
1).
c. P(0.25
X
0.78).
d. P(X = 0.41).
7. If Z is a standard normal random variable, find the value z for which:
a. P(0
Z
z) = 0.35.
b. P(–z
Z
z) = 0.142.
c. P(–z
Z
0) = 0.441.
8. Let X be a normally distributed random variable with a mean of 12 and a standard deviation of 1.5.
What proportions of the values of X are:
a. less than 14?
b. more than 8?
c. between 10 and 13?
9. The length of time patients must wait to see a doctor at an emergency room in a large hospital is
uniformly distributed between 40 minutes and 3 hours.
a. What is the probability that a patient would have to wait between 50 minutes and 2 hours?
b. What is the probability that a patient would have to wait exactly 1 hour?
c. Find the expected waiting time.
d. Find the standard deviation of the waiting time.
10. If Z is a standard normal random variable, find the value z for which:
a. the area between 0 and z is 0.4279.
b. the area to the right of z is 0.1292.
c. the area to the left of z is 0.7054.
d. the area to the left of z is 0.3156.
e. the area to the right of z is 0.8315.
f. the area between –z and z is 0.6102.
11. If X is a normal random variable with a mean of 45 and a standard deviation of 8, find the following
probabilities:
a. P(X
50).
b. P(X
32).
c. P(37
X
48).
d. P(50
X
60).
e. P(X = 45).
12.
13. A used car salesman in a small town states that, on the average, it takes him 5 days to sell a car.
Assume that the probability distribution of the length of time between sales is exponentially
distributed.
a. What is the probability that he will have to wait at least 8 days before making another sale?
b. What is the probability that he will have to wait between 6 and 10 days before making another
sale?
14. The time it takes a technician to fix a computer problem is exponentially distributed, with a mean of 15
minutes.
a. What is the probability density function for the time it takes a technician to fix a computer
problem?
b. What is the probability that it will take a technician less than 10 minutes to fix a computer
problem?
c. What is the variance of the time it takes a technician to fix a computer problem?
d. What is the probability that it will take a technician between 10 to 15 minutes to fix a computer
problem?
15. The time required to complete a particular assembly operation is uniformly distributed between 12 and
18 minutes.
a. What is the probability density function for this uniform distribution?
b. What is the probability that the assembly operation will require more than 16 minutes to complete?
c. Find the expected value and standard deviation for the assembly time.
16. If the random variable X is uniformly distributed over the interval 10
x
50, find the following
probabilities.
a. P(X
30).
b. P(X
25).
c. P(18
X
35).
d. P(X = 40).
17. If Z is a standard normal random variable, find the following probabilities.
a. P(Z
2.33).
b. P(Z
1.65).
c. P(–0.58
Z
1.58).
d. P(Z
–2.27).
18. The lifetime of a light bulb is exponentially distributed with = 0.01.
a. What are the mean and standard deviation of the light bulb’s lifetime?
b. Find the probability that a light bulb will last between 110 and 150 hours.
c. Find the probability that a light bulb will last for more than 125 hours.
19. The scores of high-school students sitting a mathematics exam were normally distributed, with a mean
of 86 and a standard deviation of 4.
a. What is the probability that a randomly selected student will have a score of 80 or higher?
b. If there were 97 680 students with scores higher than 91, how many students took the test?
20. A bank has determined that the monthly balances of the saving accounts of its customers are normally
distributed, with an average balance of $1200 and a standard deviation of $250. What proportions of
the customers have monthly balances:
a. less than $1000?
b. more than $1125?
c. between $950 and $1075?
21. The time it takes a student to complete a 3-hour business statistics sample exam paper is uniformly
distributed between 150 and 230 minutes.
a. What is the probability density function for this uniform distribution?
b. Find the probability that a student will take no more than 180 minutes to complete the sample
exam paper.
c. Find the probability that a student will take no less than 205 minutes to complete the sample exam
paper.
d. What is the expected amount of time it takes a student to complete the sample exam paper?
e. What is the standard deviation for the amount of time it takes a student to complete the sample
exam paper?
22. A continuous random variable X has the probability density function f(x) = –0.02x + 0.2, 0
x
10.
Find the following probabilities.
a. P(X
7).
b. P(X
8).
c. P(X = 9).
Hint: P(a
x
b) =
23. If Z is a standard normal random variable, find the value z for which:
a. P(0
Z
z) = 0.276.
b. P(Z
z) = 0.341.
c. P(Z
z) = 0.819.
d. P(–z
Z
z) = 0.785.
e. P(Z
z) = 0.9279.
24. A supermarket receives a delivery each morning at a time that varies uniformly between 5:00 and
7:00am.
a. Find the probability that the delivery on a given morning will occur between 5:30 and 5:45am.
b. What is the expected time of delivery?
c. Determine the standard deviation of the delivery time.
d. Find the probability that the time of delivery will be within half a standard deviation of the
expected time.
25. A continuous random variable X has the probability density function f(x) = 2
x
e2−
, x 0.
a. Find the mean and standard deviation of X.
b. What is the probability that X is between 1 and 3?
c. What is the probability that X is at most 2?
26. The recent average starting salary for new college graduates in computer information systems is
$47 500. Assume that salaries are normally distributed, with a standard deviation of $4500.
a. What is the probability of a new graduate receiving a salary between $45 000 and $50 000?
b. What is the probability of a new graduate getting a starting salary in excess of $55 000?
c. What percentage of starting salaries are no more than $42 250?
d. What is the cutoff for the bottom 5% of the salaries?
e. What is the cutoff for the top 3% of the salaries?
27. In a shopping centre, the waiting time for an elevator is found to be uniformly distributed between 1
and 5 minutes.
a. What is the probability density function for this uniform distribution?
b. What is the probability of waiting no more than 3 minutes?
c. What is the probability that the elevator arrives in the first 30 seconds?
d. What is the probability of a waiting time between 2 and 3 minutes?
e. What is the expected waiting time?
28. The active lifetime of laptop computers is normally distributed, with a mean of 36 months and a
standard deviation of 6 months.
a. What is the probability that a randomly selected laptop will last more than 3.5 years?
b. What proportion of the laptops will last less than 32 months?
c. What proportion of the laptops will last between 2 and 3 years?
29. The weights of cans of soup produced by a company are normally distributed, with a mean of 150 g
and a standard deviation of 5 g.
a. What is the probability that a can of soup selected randomly from the entire production will weigh
at most 143 g?
b. Determine the minimum weight of the heaviest 5% of all cans of soup produced.
c. If 28 390 of the cans of soup of the entire production weigh at least 157.5 g, how many cans of
soup have been produced?
30. Consider a binomial random variable X with n = 300 and p = 0.02. Approximate the values of the
following probabilities.
a. P(X = 4).
b. P(X 5).
c. P(X 8).
31. Let X be a binomial random variable with n = 25 and p = 0.6. Approximate the following probabilities,
using the normal distribution.
a. P(X
20).
b. P(X
15).
c. P(X = 10).
32. Suppose it is known that 60% of students at a particular university are smokers. A sample of 500
students from the university is selected at random. Approximate the probability that at least 280 of
these students are smokers.
33. Let X be a binomial random variable with n = 100 and p = 0.7. Approximate the following
probabilities, using the normal distribution.
a. P(X = 75).
b. P(X 70).
c. P(X 60).
34. A fair coin is tossed 500 times. Approximate the probability that the number of heads observed is
between 240 and 270 (inclusive).
35. The publisher of a daily newspaper claims that 90% of its subscribers are under the age of 30. Suppose
that a sample of 300 subscribers is selected at random. Assuming the claim is correct, approximate the
probability of finding at least 240 subscribers in the sample under the age of 30.
