CHAPTER 8: CONTINUOUS PROBABILITY DISTRIBUTIONS
TRUE/FALSE
1. Since there is an infinite number of values a continuous random variable can assume, the probability of
each individual value is virtually 0.
2. A continuous probability distribution represents a random variable having an infinite number of
outcomes which may assume any number of values within an interval.
3. Continuous probability distributions describe probabilities associated with random variables that are
able to assume any finite number of values along an interval.
4. A continuous random variable X has a uniform distribution between 10 and 20 (inclusive), then the
probability that X falls between 12 and 15 is 0.30.
5. A continuous random variable is one that can assume an uncountable number of values.
6. A continuous random variable X has a uniform distribution between 5 and 15 (inclusive), then the
probability that X falls between 10 and 20 is 1.0.
7. A continuous random variable X has a uniform distribution between 5 and 25 (inclusive), then P(X =
15) = 0.05.
8. We distinguish between discrete and continuous random variables by noting whether the number of
possible values is countable or uncountable.
9. In practice, we frequently use a continuous distribution to approximate a discrete one when the number
of values the variable can assume is countable but large.
10. Let X represent weekly income expressed in dollars. Since there is no set upper limit, we cannot
identify (and thus cannot count) all the possible values. Consequently, weekly income is regarded as a
continuous random variable.
11. To be a legitimate probability density function, all possible values of f(x) must be non-negative.
12. To be a legitimate probability density function, all possible values of f(x) must lie between 0 and 1
(inclusive).
13. The sum of all values of f(x) over the range of [a, b] must equal one.
14. A probability density function shows the probability for each value of X.
15. If X is a continuous random variable on the interval [0, 10], then P(X > 5) = P(X 5).
16. If X is a continuous random variable on the interval [0, 10], then P(X = 5) = f(5) = 1/10.
17. If a point y lies outside the range of the possible values of a random variable X, then f(y) must equal
zero.
MULTIPLE CHOICE
1. Which of the following is always true for all probability density functions of continuous random
variables?
a.
The probability at any single point is zero.
b.
They contain an uncountable number of possible values.
c.
The total area under the density function f(x) equals 1.
d.
All of these choices are true.
2. The probability density function, f(x), for any continuous random variable X, represents:
a.
all possible values that X will assume within some interval a x b.
b.
the probability that X takes on a specific value x.
c.
the height of the density function at x.
d.
None of these choices.
3. Which of the following represents a difference between continuous and discrete random variables?
a.
Continuous random variables assume an uncountable number of values, and discrete
random variables do not.
b.
The probability for any individual value of a continuous random variable is zero, but for
discrete random variables it is not.
c.
Probability for continuous random variables means finding the area under a curve, while
for discrete random variables it means summing individual probabilities.
d.
All of these choices are true.
4. Suppose f(x) = 0.25. What range of possible values can X take on and still have the density function be
legitimate?
a.
[0, 4]
c.
[−2, +2]
b.
[4, 8]
d.
All of these choices are true.
5. What is the shape of the probability density function for a uniform random variable on the interval
[a, b]?
a.
A rectangle whose X values go from a to b.
b.
A straight line whose height is 1/(b − a) over the range [a, b].
c.
A continuous probability density function with the same value of f(x) from a to b.
d.
All of these choices are true.
6. Which of the following is true about f(x) when X has a uniform distribution over the interval [a, b]?
a.
The values of f(x) are different for various values of the random variable X.
b.
f(x) equals one for each possible value of X.
c.
f(x) equals one divided by the length of the interval from a to b.
d.
None of these choices.
7. Suppose f(x) = 1/4 over the range a x b, and suppose P(X > 4) = 1/2. What are the values for a and
b?
a.
0 and 4
b.
2 and 6
c.
Can be any range of x values whose length (b − a) equals 4.
d.
Cannot answer with the information given.
8. The probability density function f(x) for a uniform random variable X defined over the interval [2, 10]
is
a.
0.20
c.
4
b.
8
d.
None of these choices.
9. If the random variable X has a uniform distribution between 40 and 50, then P(35 X 45) is:
a.
1.0
c.
0.1
b.
0.5
d.
undefined.
10. The probability density function f(x) of a random variable X that has a uniform distribution between a
and b is
a.
(b + a)/2
c.
(a − b)/2
b.
1/b − 1/a
d.
None of these choices.
11. Which of the following does not represent a continuous uniform random variable?
a.
f(x) = 1/2 for x between −1 and 1, inclusive.
b.
f(x) = 10 for x between 0 and 1/10, inclusive.
c.
f(x) = 1/3 for x = 4, 5, 6.
d.
None of these choices represents a continuous uniform random variable.
COMPLETION
1. A(n) ____________________ random variable is one that assumes an uncountable number of possible
values.
2. For a continuous random variable, the probability for each individual value of X is
____________________.
3. Probability for continuous random variables is found by finding the ____________________ under a
curve.
4. A(n) ____________________ random variable has a density function that looks like a rectangle and
you can use areas of a rectangle to find probabilities for it.
5. Suppose X is a continuous random variable for X between a and b. Then its probability
____________________ function must non-negative for all values of X between a and b.
6. The total area under f(x) for a continuous random variable must equal ____________________.
7. The probability density function of a uniform random variable on the interval [0, 5] must be
____________________ for 0 x 5.
8. To find the probability for a uniform random variable you take the ____________________ times the
____________________ of its corresponding rectangle.
9. You can use a continuous random variable to ____________________ a discrete random variable that
takes on a countable, but very large, number of possible values.
SHORT ANSWER
1. A continuous random variable X has the following probability density function:
f(x) = 1/4, 0 x 4
Find the following probabilities:
a.
P(X 1)
b.
P(X 2)
c.
P(1 X 2)
d.
P(X = 3)
ANS:
b.
0.50
0.25
Waiting Time
The length of time patients must wait to see a doctor at an emergency room in a large hospital has a
uniform distribution between 40 minutes and 3 hours.
2. {Waiting Time Narrative} What is the probability density function for this uniform distribution?
3. {Waiting Time Narrative} What is the probability that a patient would have to wait between one and
two hours?
4. {Waiting Time Narrative} What is the probability that a patient would have to wait exactly one hour?
ANS:
5. {Waiting Time Narrative} What is the probability that a patient would have to wait no more than one
hour?
6. The time required to complete a particular assembly operation has a uniform distribution between 25
and 50 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 40 minutes to
complete?
c.
Suppose more time was allowed to complete the operation, and the values of X were
extended to the range from 25 to 60 minutes. What would f(x) be in this case?
b.
0.40
7. Suppose f(x) equals 1/50 on the interval [0, 50].
a.
What is the distribution of X?
b.
What does the graph of f(x) look like?
c.
Find P(X 25)
d.
Find P(X 25)
e.
Find P(X = 25)
f.
Find P(0 < X < 3)
g.
Find P(−3 < X < 0)
h.
Find P(0 < X < 50)
a.
b.
c.
0.50
d.
0.50
e.
0
Electronics Test
The time it takes a student to finish a electronics test has a uniform distribution between 50 and 70
minutes.
8. {Electronics Test Narrative} What is the probability density function for this uniform distribution?
9. {Electronics Test Narrative} Find the probability that a student will take more than 60 minutes to
finish the test.
10. {Electronics Test Narrative} Find the probability that a student will take no less than 55 minutes to
finish the test.
11. {Electronics Test Narrative} Find the probability that a student will take exactly one hour to finish the
test.
12. {Electronics Test Narrative} What is the median amount of time it takes a student to finish the test?
13. {Electronics Test Narrative} What is the mean amount of time it takes a student to finish the test?
Subway Waiting Time
At a subway station the waiting time for a subway is found to be uniformly distributed between 1 and
5 minutes.
14. {Subway Waiting Time Narrative} What is the probability density function for this uniform
distribution?
15. {Subway Waiting Time Narrative} What is the probability of waiting no more than 3 minutes?
16. {Subway Waiting Time Narrative} What is the probability that the subway arrives in the first minute
and a half?
17. {Subway Waiting Time Narrative} What is the median waiting time for this subway?
18. A national standardized testing company can tell you your relative standing on an exam without
divulging the mean or the standard deviation of the exam scores.
19. If your golf score is 3 standard deviations below the mean, its corresponding value on the Z
distribution is −3.
20. If we standardize the normal curve, we express the original X values in terms of their number of
standard deviations away from the mean.
21. A normal distribution is symmetric; therefore the probability of being below the mean is 0.50 and the
probability of being above the mean is 0.50.
22. A random variable X is standardized by subtracting the mean and dividing by the variance.
23. A random variable X has a normal distribution with mean 132 and variance 36. If x = 120, its
corresponding value of Z is 2.0.
24. A random variable X has a normal distribution with a mean of 250 and a standard deviation of 50.
Given that X = 175, its corresponding value of Z is −1.50.
25. Given that Z is a standard normal random variable, a negative value of Z indicates that the standard
deviation of Z is negative.
26. In the standard normal distribution, z0.05 = 1.645 means that 5% of all values of z are below 1.645 and
95% are above it.
27. The probability that a standard normal random variable Z is less than −3.5 is approximately 0.
28. If the value of Z is z = 99, that means you are at the 99th percentile on the Z distribution.
29. The 10th percentile of a Z distribution has 10% of the Z-values lying above it.
30. The probability that Z is less than −2 is the same as one minus the probability that Z is greater than +2.
31. Suppose X has a normal distribution with mean 70 and standard deviation 5. The 50th percentile of X
is 70.
32. Which of the following is not a characteristic for a normal distribution?
a.
It is symmetrical.
b.
The mean is always zero.
c.
The mean, median, and mode are all equal.
d.
It is a bell-shaped distribution.
33. If X has a normal distribution with mean 60 and standard deviation 6, which value of X corresponds
with the value z = 1.96?
a.
x = 71.76
b.
x = 67.96
c.
x = 61.96
d.
x = 48.24
34. A standard normal distribution is a normal distribution with:
a.
a mean of zero and a standard deviation of one.
b.
a mean of one and a standard deviation of zero.
c.
a mean always larger than the standard deviation.
d.
None of these choices.
35. What proportion of the data from a normal distribution is within two standard deviations from the
mean?
a.
0.3413
b.
0.4772
c.
0.6826
d.
0.9544
36. Given that Z is a standard normal random variable, the area to the left of a value z is expressed as
a.
P(Z z)
b.
P(Z z)
c.
P(0 Z z)
d.
P(Z −z)
37. Given that Z is a standard normal variable, the variance of Z:
a.
is always greater than 2.0.
b.
is always greater than 1.0.
c.
is always equal to 1.0.
d.
cannot assume a specific value.
38. Given that Z is a standard normal random variable, a negative value (z) on its distribution would
indicate:
a.
z is to the left of the mean.
b.
the standard deviation of this Z distribution is negative.
c.
the area between zero and the value z is negative.
d.
None of these choices.
39. A larger standard deviation of a normal distribution indicates that the distribution becomes:
a.
narrower and more peaked.
b.
flatter and wider.
c.
more skewed to the right.
d.
more skewed to the left.
40. In its standardized form, the normal distribution:
a.
has a mean of 0 and a standard deviation of 1.
b.
has a mean of 1 and a variance of 0.
c.
has an area equal to 0.5.
d.
cannot be used to approximate discrete probability distributions.
41. Most values of a standard normal distribution lie between:
a.
0 and 1
b.
−3 and 3
c.
0 and 3
d.
minus infinity and plus infinity
42. Stacy took a math test whose mean was 70 and standard deviation was 5. The total points possible was
100. Stacey’s results were reported to be at the 95th percentile. What was Stacey’s actual exam score,
rounded to the nearest whole number?
a.
95
b.
78
c.
75
d.
62
43. Tanner took a statistics test whose mean was 80 and standard deviation was 5. The total points
possible was 100. Tanner’s score was 2 standard deviations below the mean. What was Tanner’s score,
rounded to the nearest whole number?
a.
78
b.
70
c.
90
d.
None of these choices.
44. Lamont took a psychology exam whose mean was 70 with standard deviation 5. He also took a
calculus exam whose mean was 80 with standard deviation 10. He scored 85 on both exams. On which
exam did he do better compared to the other students who took the exam?
a.
He did better on the psychology exam, comparatively speaking.
b.
He did better on the calculus exam, comparatively speaking.
c.
He did the same on both exams, relatively speaking.
d.
Cannot tell without more information.
45. Suppose Lamont’s exam score was at the 80th percentile on an exam whose mean was 90. What was
Lamont’s exam score?
a.
76.81
b.
72.00
c.
80.00
d.
Cannot tell without more information.
46. Suppose X has a normal distribution with mean 40 and standard deviation 2. Shifting all the X values
to the right 10 units results in a normal distribution with mean ____________________ and standard
deviation ____________________.
47. ____________________ the value of
in a normal distribution will make it wider.
48. We standardize a random variable by subtracting its ____________________ and dividing by its
____________________.
49. Suppose X has a normal distribution with mean 10 and standard deviation 2. The probability that X is
less than 8 is equal to the probability that Z is less than ____________________.
50. P(Z > 1.9) = ____________________ P(Z < 1.9).
51. P(1 < Z < 2) = P(Z < 2) − ____________________.
52. The mean of the standard normal distribution is ____________________ and the standard deviation is
____________________.
53. P(Z > 3.00) is approximately ____________________.
54. P(Z < 3.00) is approximately ____________________.
55. Suppose X is a normal random variable with mean 70 and standard deviation 3. Then P(X = 3) =
____________________.
56. Z.025 is the value of Z such that the area to the ____________________ of Z is .9750.
Battery Life
A certain brand of batteries has a lifetime that has a normal distribution with a mean of 3,750 hours
and a standard deviation of 300 hours.
57. {Battery Life Narrative} What proportion of these batteries will last for more than 4,000 hours?
58. {Battery Life Narrative} What proportion of these batteries will last less than 3,600 hours?
ANS:
59. {Battery Life Narrative} What proportion of these batteries will last between 3,800 and 4,100 hours?
ANS:
60. {Battery Life Narrative} What lifetime should the manufacturer advertise for these batteries in order
that only 2% of the lamps will wear out before the advertised lifetime?
ANS:
Diet
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 10 pounds and a standard deviation of 5
pounds.
61. {Diet Narrative} What proportion of the dieters lost more than 12 pounds?
62. {Diet Narrative} What proportion of the dieters gained weight?
63. {Diet Narrative} If a dieter is selected at random, what is the probability that the dieter lost more than
7.5 pounds?
64. 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
ANS: