Chapter 8—Continuous probability distributions
MULTIPLE CHOICE
1. Given that Z is a standard normal random variable, P(–1.3 Z 1.8) is:
A.
0.4032.
B.
0.5248.
C.
0.8673.
D.
0.4641.
2. Given that Z is a standard normal variable, the value z for which P(Z z) = 0.2580 is:
A.
0.70.
B.
0.758.
C.
–0.65.
D.
0.242.
3. If the random variable X is exponentially distributed with parameter
= 3, then P(X
2), up to 4
decimal places, is:
A.
0.3333.
B.
0.5000.
C.
0.6667.
D.
0.0025.
4. If the random variable X is exponentially distributed with parameter = 1.75, then P(1.5 X 3.8), up
to 4 decimal places, is:
A.
0.0711.
B.
0.0473.
C.
0.1184.
D.
0.4739.
5. If X is a normal random variable with a standard deviation of 10, then 3X has a standard deviation
equal to:
A.
10.
B.
13.
C.
30.
D.
90.
6. Given that X is a normal variable, which of the following statements is (are) true?
A.
The variable X + 5 is also normally distributed.
B.
The variable X – 5 is also normally distributed.
C.
The variable 5X is also normally distributed.
D.
All of the above statements are true.
7. If the random variable X is exponentially distributed with parameter
= 4, then the probability P(X
0.25), up to 4 decimal places, is:
A.
0.6321.
B.
0.3679.
C.
0.2500.
D.
None of the above answers is correct.
8. 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 usually larger than the standard deviation.
D.
a mean always larger than the standard deviation.
9. If Z is a standard normal random variable, then P(–2.18 Z –0.53) is:
A.
0.6873.
B.
0.2835.
C.
0.4854.
D.
0.2019.
10. If Z is a standard normal random variable, then the value z for which P(–z Z z) equals 0.4778 is:
A.
0.4011.
B.
0.6400.
C.
0.8789.
D.
0.2389.
11. If the continuous random variable X is uniformly distributed over the interval [15, 25], then the mean
of X is:
A.
40.
B.
25.
C.
20.
D.
15.
12. If the random variable X is normally distributed with a mean of 75 and a standard deviation of 8, then
P(X
75) is:
A.
0.125.
B.
0.500.
C.
0.625.
D.
0.975.
13. Given that Z is a standard normal random variable, what is the value of Z if the area to the right of Z is
0.1949?
A.
0.51.
B.
–0.51.
C.
0.86.
D.
–0.86.
14. Given that Z is a standard normal random variable, what is the value of Z if the area to the right of Z is
0.8212?
A.
0.92.
B.
–0.05.
C.
0.05.
D.
–0.92.
15. Given that Z is a standard normal random variable, P(Z > 2.68) is:
A.
0.0037.
B.
0.5037.
C.
0.9963.
D.
0.4963.
16. Given that the random variable X is normally distributed with a mean of 20 and a standard deviation of
7, P(3 X 28) is:
A.
0.4925.
B.
0.3729.
C.
0.8654.
D.
0.1196.
17. Which of the following is always true for all probability density functions of continuous random
variables?
A.
They are symmetrical.
B.
They are bell-shaped.
C.
The area under the curve is 1.0.
D.
They have the same height.
18. What proportion of the data from a normal distribution is within 3 standard deviations of the mean?
A.
0.3413.
B.
0.4772.
C.
0.6826.
D.
0.9544.
19. Which of the following distributions is suitable to model the length of time that elapses before the first
telephone call is received by a switchboard?
A.
Exponential.
B.
Normal.
C.
Poisson.
D.
Uniform.
20.Like the normal distribution, the exponential density function f(x):
A.
is bell-shaped.
B.
is symmetrical.
C.
approaches infinity as x approaches zero.
D.
approaches zero as x approaches infinity.
21. The mean of the exponential distribution equals the mean of the Poisson distribution only when the
former distribution has a mean equal to:
A.
1.0.
B.
0.50.
C.
0.25.
D.
any value smaller than 1.0.
22. Which of the following distributions is appropriate to measure the length of time between arrivals at a
grocery checkout counter?
A.
Uniform.
B.
Normal.
C.
Exponential.
D.
Poisson.
23. 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).
24. Which of the following distributions are symmetrical?
A.
Exponential and uniform.
B.
Uniform and normal.
C.
Normal and exponential.
D.
All continuous distributions are symmetrical.
25. If the z-value for a given value x of the random variable X is z = 2.326, and the distribution of X is
normal with a mean of 50 and a standard deviation of 5, to what x-value does this z–value correspond?
A.
11.63.
B.
61.63.
C.
63.16.
D.
16.31.
26. If the random variable X is exponentially distributed, then the mean of X will be:
A.
greater than the median.
B.
smaller than the median.
C.
the same as the median.
D.
None of the above answers is correct.
27. If Z is a standard normal random variable, the area between z = 0.0 and z =1.30 is 0.4032, while the
area between z = 0.0 and z = 1.50 is 0.4332. What is the area between z = –1.30 and z = 1.50?
A.
0.0300.
B.
0.0668.
C.
0.0968.
D.
0.8364.
28. If Z is a standard normal random variable, the area between z = –1.68 and z = –1.28, compared to the
area between z = 1.28 and z = 1.68, will be:
A.
the same.
B.
larger.
C.
smaller.
D.
None of the above is correct.
29. The probability that a continuous random variable X will assume any specific value is:
A.
0.0.
B.
1.0.
C.
0.50.
D.
any value between –0.5 and 0.5.
30. The expected value, E(X), of a uniform random variable X defined over the interval
bxa
, is:
A.
a + b.
B.
a – b.
C.
(a + b)/2.
D.
(a – b)/2.
31. 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 area under the curve at x.
D.
the height of the function at x.
32. The height of the function for a uniform probability density function f(x):
A.
is different for various values of the random variable X.
B.
is the same for various values of the random variable X.
C.
increases as the values of the random variable X increase.
D.
None of the above statements is correct.
33. Which of the following is not true for a random variable X that is uniformly distributed over the
interval
bxa
?
A.
E(X) = (a + b)/2.
B.
V(X) =
/12.
C.
= (b – a)/6.
D.
if a x b.
34.The function f(x) that defines the probability distribution of a continuous random variable X is a:
A.
binomial function.
B.
normal function.
C.
Poisson function.
D.
probability density function.
35.The probability density function f(x) for a uniform random variable X defined over the interval [1, 11] is:
A.
any value between 1 and 11.
B.
0.100.
C.
0.091.
D.
above 11.
36. If the random variable X is uniformly distributed between 40 and 60, then P(35 X 45) is:
A.
1.0.
B.
0.5.
C.
0.25.
D.
0.0.
37. Which of the following is not true for a normal distribution?
A.
It is unimodal.
B.
It is symmetrical.
C.
It is discrete.
D.
It has a bell shape.
38. Which of the following distributions is considered the cornerstone distribution of statistical inference?
A.
Binomial distribution.
B.
Normal distribution.
C.
Poisson distribution.
D.
Uniform distribution.
39. The probability density function f(x) of a random variable X that is normally distributed is completely
determined once the:
A.
mean and median of X are specified.
B.
median and mode of X are specified.
C.
mean and mode of X are specified.
D.
mean and variance of X are specified.
40. The probability density function f(x) of a random variable X that is uniformly distributed between a
and b is:
A.
1/(b – a).
B.
1/(a – b).
C.
(b – a)/2.
D.
(a – b)/2.
41. Which of the following is not a characteristic of a normal distribution?
A.
It is a symmetrical distribution.
B.
The mean is always zero.
C.
The mean, median and mode are all equal.
D.
It is a bell-shaped distribution.
42. 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.
43. Given that Z is a standard normal random variable, a positive z value means that:
A.
the value z is to the left of the mean.
B.
the value z is to the right of the median.
C.
the z value is to the right of the mean.
D.
both B and C are correct.
44. Given that Z is a standard normal random variable, the mean of Z is:
A.
smaller than the median.
B.
larger than the mode.
C.
always equal to zero.
D.
always smaller than zero.
45. Which of the following is not true for an exponential distribution with parameter
?
A.
/1=
B.
/1=
C.
The distribution is completely determined once the value of
is known.
D.
The distribution is a two-parameter distribution, since the mean and standard deviation are
equal.
46. A smaller standard deviation of a normal distribution indicates that the distribution becomes:
A.
more skewed to the left.
B.
flatter and wider.
C.
narrower and more peaked.
D.
symmetrical.
47. Given that Z is a standard normal variable, the value z for which P(Z
z) = 0.6736 is:
A.
0.1736.
B.
0.45.
C.
–0.1736.
D.
–0.45.
48. If the mean of an exponential distribution is 2, then the value of the parameter
is:
A.
4.
B.
2.
C.
1.
D.
0.5.
49. Given that X is a binomial random variable, the binomial probability P(X
x) is approximated by the
area under a normal curve to the right of:
A.
x – 0.5.
B.
x + 0.5.
C.
x – 1.
D.
x + 1.
50. Given a binomial distribution with n trials and probability p of a success on any trial, a conventional
rule of thumb is that the normal distribution will provide an adequate approximation of the binomial
distribution if:
A.
np
5 and n(1–p)
5.
B.
np
5 and n(1–p)
5.
C.
np
5 and n(1–p)
5.
D.
np
5 and n(1–p)
5.
51. Suppose that the probability p of a success on any trial of a binomial distribution equals 0.80. For
which value of the number of trials, n, would the normal distribution provide a good approximation to
the binomial distribution?
A.
25.
B.
20.
C.
10.
D.
15.
52. Given that X is a binomial random variable, the binomial probability P(X x) is approximated by the
area under a normal curve to the left of:
A.
x.
B.
–x.
C.
x + 0.5.
D.
x – 0.5.
TRUE/FALSE
1. A continuous probability distribution represents a random variable having an infinite number of
outcomes that may assume any number of values within an interval.
2. If we standardise the normal curve, we express the original x values in terms of their number of
standard deviations away from the mean.
3. In the exponential distribution, the value of x can be any of an infinite number of values in the given
range.
4. In the normal distribution, the mean, median and mode are all at the same position on the horizontal
axis since the distribution is symmetric.
5. In the normal distribution, the curve is asymptotic but never intercepts the horizontal axis either to the
left or right.
6. In the normal distribution, the flatter the curve, the smaller the standard deviation
7. In the normal distribution, the total area under the curve is equal to one.
8. In the normal distribution, the right half of the curve is slightly larger than the left half.
9. Continuous probability distributions describe probabilities associated with random variables that are
able to assume any of a finite number of values along an interval.
10. A random variable X is standardised when each value of X has the mean of X subtracted from it, and
the difference is divided by the standard deviation of X.
11. Using the standard normal curve, the area between z = 0 and z = 3.50 is about 0.50.
12. Using the standard normal curve, the probability or area between z = –1.28 and z = 1.28 is 0.7994.
13. Let z1 be a z-score that is unknown but identifiable by position and area. If the area to the right of z1 is
0.7291, the value of z1 is –0.61.
14. Let z1 be a z-score that is unknown but identifiable by position and area. If the symmetrical area
between –z1 and + z1 is 0.9544, the value of z1 is 2.0.
15. Using the standard normal curve, the z-score representing the 10th percentile is 1.28.
16. Using the standard normal curve, the z-score representing the 75th percentile is 0.67.
17. Using the standard normal curve, the z-score representing the 90th percentile is 1.28.
18. The mean and standard deviation of a normally distributed random variable that has been standardised
are one and zero, respectively.
19. The mean and standard deviation of an exponential random variable are equal to each other.
20. A random variable X is normally distributed with a mean of 150 and a variance of 25. Given that
X = 120, its corresponding z-score is 6.0.
21. A random variable X is normally distributed with a mean of 250 and a standard deviation of 50. Given
that X = 175, its corresponding z-score is –1.50.
22. For a normal curve, if the mean is 25 minutes and the standard deviation is 5 minutes, the area to the
right of 25 minutes is 0.50.
23. For a normal curve, if the mean is 20 minutes and the standard deviation is 5 minutes, the area to the
right of 13 minutes is 0.9192.
24. If the random variable X is exponentially distributed with
= 2 parameter, then the variance of the
distribution is 0.5.