Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Decide whether or not the function is a probability density function on the indicated interval.
1)
f(x) =3x2
16 ; [–2, 2]
1)
A)
Yes
B)
No
Find the proportion of observations of a standard normal distribution that are between the given z–scores.
2)
0 and 2.5
2)
A)
0.9940
B)
0.4062
C)
0.9938
D)
0.4938
Find the expected value, the variance, and the standard deviation, when they exist, for the probability density function.
Give an exact answer for the expected value and round the variance and standard deviation to four decimal places when
appropriate.
3)
f(x) =
x3
10 if 0
x 3
19
3x3if x >3
3)
A)
3137
450 ; 83.8385; does not exist
B)
3137
450 ; does not exist; does not exist
C)
3137
450 ; 83.8385; 9.1563
D)
3137
450 ; 9.1563; does not exist
Solve the problem.
4)
The annual rainfall in Maine is a random variable with probability density function defined by
f(x) =1
15 x +1
2 for x in [0, 5]. Find the mean annual rainfall.
4)
A)
3.000
B)
3.194
C)
3.360
D)
2.994
Find the expected value of the probability density function to the nearest hundredth.
5)
f(x) =3x2
98 ; [3, 5]
5)
A)
4.20
B)
1.31
C)
4.16
D)
4.78
D)
Solve the problem.
6)
The number of new mini–vans sold by a particular salesperson during the month of March is
exponentially distributed with a mean of 3. What is the probability that the salesperson will sell
between 2 and 6 mini–vans in March?
6)
A)
0.378
B)
0.245
C)
0.284
D)
0.349
D)
Find a value of k that will make f a probability density function on the indicated interval.
7)
f(x) = kx; [0, 6]
7)
A)
1
18
B)
1
35
C)
1
9
D)
1
36
D)
Find the probability to the nearest hundredth that the value of the random variable is within one standard deviation of
the mean.
8)
f(x) = 2(1 – x); [0, 1]
8)
A)
0.64
B)
0.50
C)
0.75
D)
0.82
D)
Solve the problem.
9)
A machine fills quart soda bottles with an average of 32.3 oz per bottle, with a standard deviation
of 1.2 oz. What is the probability that a filled bottle will contain less than 32 oz? Assume the
distribution is normal.
9)
A)
60%
B)
40%
C)
38%
D)
41%
Decide whether or not the function is a probability density function on the indicated interval.
10)
f(x) =3
98 x2; [3, 5]
10)
A)
No
B)
Yes
11)
f(x) = 3x2; [–2, 2]
11)
A)
No
B)
Yes
Find the mean and standard deviation of the specified probability density function.
12)
f(x) = 0.3e–0.3x for [0, )
12)
A)
µ= 33.33, = 33.33
B)
µ= 3.33, = 3.33
C)
µ= 3.75, = 3.75
D)
µ= 3.33, = 3.50
Decide whether or not the function is a probability density function on the indicated interval.
13)
f(x) =4
74 x3; [0, 3]
13)
A)
Yes
B)
No
Find the variance of the probability density function to the nearest hundredth.
14)
f(x) = 4x–5; [1, )
14)
A)
0.20
B)
0.22
C)
0.25
D)
0.33
Find the standard deviation of the probability density function to the nearest hundredth.
15)
f(x) = 1 –1
x; [1, 4]
15)
A)
0.76
B)
0.71
C)
0.75
D)
0.73
Find a value of k that will make f a probability density function on the indicated interval.
16)
f(x) = kx2; [–1, 4]
16)
A)
1
16
B)
3
65
C)
1
21
D)
3
64
Decide whether or not the function is a probability density function on the indicated interval.
17)
f(x) = 2x; [–2, 5]
17)
A)
Yes
B)
No
Find the mean and standard deviation of the specified probability density function.
18)
f(x) = 0.2e–0.2x for [0, )
18)
A)
µ= 5, = 5
B)
µ= 0.5, = 0.6
C)
µ= 0.5, = 0.5
D)
µ= 6, = 6
Solve the problem.
19)
A company installs 5000 light bulbs. Each bulb has an average life of 500 hours with a standard
deviation of 100 hours. The life of each bulb is approximated by a normal curve. Find the number
of bulbs that can be expected to last less than 500 hours.
19)
A)
3000
B)
2500
C)
2400
D)
1000
Find the standard deviation of the probability density function to the nearest hundredth.
20)
f(x) = 2(1 – x); [0, 1]
20)
A)
0.31
B)
0.25
C)
0.24
D)
0.33
Solve the problem.
21)
The annual rainfall in Maine is a random variable with probability density function defined by
f(x) =1
15 x +1
2 for x in [0, 5]. Find the probability of rainfall less than two standard deviations
below the mean.
21)
A)
0.037
B)
0.026
C)
0
D)
0.031
22)
The annual rainfall in Maine is a random variable with probability density function defined by
f(x) =1
15 x +1
2 for x in [0, 5]. Find the probability that the rainfall is greater than the mean.
22)
A)
0.512
B)
0.500
C)
0.553
D)
0.549
Find the standard deviation of the probability density function to the nearest hundredth.
23)
f(x) =1
5; [0, 5]
23)
A)
1.45
B)
1.40
C)
1.02
D)
1.44
Find the expected value of the probability density function to the nearest hundredth.
24)
f(x) =1
6; [2, 8]
24)
A)
4.50
B)
5.33
C)
5.00
D)
6.50
Solve the problem.
25)
The life (in years) of a certain species of whale is a random variable with probability density
function defined by f(x) =1
91+2
xfor x in [4, 9]. Find the mean life of this species of whale.
25)
A)
6.4 years
B)
8.1 years
C)
7.8 years
D)
9.6 years
Use the standard normal curve table to find the z–score for the given condition.
26)
30.15% of the total area is to the left of z.
26)
A)
–0.88
B)
0.52
C)
–0.52
D)
–0.53
Find the mean and standard deviation of the specified probability density function.
27)
f(x) = 4 for [2.25, 2.50]
27)
A)
µ= 2.375, = 0.144
B)
µ= 2.500, = 0.069
C)
µ= 2.500, = 0.070
D)
µ= 2.375, = 0.072
Find the proportion of observations of a standard normal distribution that are between the given z–scores.
28)
–0.75 and 0.75
28)
A)
0.5528
B)
0.2734
C)
0.7734
D)
0.5467
Find the expected value of the probability density function to the nearest hundredth.
29)
f(x) =1
3; [3, 6]
29)
A)
4.17
B)
4.00
C)
5.00
D)
4.50
Find a value of k that will make f a probability density function on the indicated interval.
30)
f(x) = kx3; [0, 2]
30)
A)
1
8
B)
1
4
C)
1
15
D)
4
15
Solve the problem.
31)
The life span of a certain insect in days is uniformly distributed over the interval [20, 36]. What is
the standard deviation?
31)
A)
5.774
B)
4.33
C)
4.619
D)
10.392
Find the variance of the probability density function to the nearest hundredth.
32)
f(x) = 1 –1
x; [1, 4]
32)
A)
0.52
B)
0.61
C)
0.57
D)
0.53
Find the proportion of observations of a standard normal distribution that are between the given z–scores.
33)
0 and 0.94
33)
A)
0.8264
B)
0.3264
C)
0.1736
D)
0.3289
Find the median of the random variable for the given probability density function.
34)
f(x) =x
8–1
2; [4, 8]
34)
A)
6
B)
6.83
C)
5.41
D)
0.50
Find a value of k that will make f a probability density function on the indicated interval.
35)
f(x) = kx; [0, 5]
35)
A)
1
25
B)
2
25
C)
1
5
D)
2
5
Solve the problem.
36)
A company installs 5000 light bulbs. Each bulb has an average life of 500 hours with a standard
deviation of 100 hours. The life of each bulb is approximated by a normal curve. Find the number
of bulbs that can be expected to last between 290 hours and 540 hours.
36)
A)
1641
B)
3188
C)
1639
D)
3190
Decide whether or not the function is a probability density function on the indicated interval.
37)
f(x) =3
63 x2; [1, 4]
37)
A)
Yes
B)
No
Solve the problem.
38)
The mean clotting time of blood is 7.35 seconds, with a standard deviation of 0.35 seconds. What is
the probability that blood clotting time will be less than 7 seconds? Assume the distribution is
normal.
38)
A)
14%
B)
84%
C)
15%
D)
16%
Decide whether or not the function is a probability density function on the indicated interval.
39)
f(x) =1
3x –1
6; [3, 4]
39)
A)
No
B)
Yes
Find the proportion of observations of a standard normal distribution that are between the given z–scores.
40)
0 and 0.75
40)
A)
0.4599
B)
0.9599
C)
0.2734
D)
0.4591
Find the probability to the nearest hundredth that the random variable of the probability density function has a value
greater than the mean.
41)
f(x) =1
6; [2, 8]
41)
A)
0.45
B)
0.48
C)
0.50
D)
0.53
Solve the problem.
42)
The time between major earthquakes in a particular region of the Mediterranean is a random
variable with probability density function f(x) =1
1800 e–x/1800 for x in [0, ), where t is measured
in days. Find the expected value and the standard deviation of this probability density function.
42)
A)
µ=3600 days; =2545.2 days
B)
µ=1800 days; =1800 days
C)
µ=1800 days; =2545.2 days
D)
µ=1800 days; =3600 days
Find the probability to the nearest hundredth that the random variable of the probability density function has a value
greater than the mean.
43)
f(x) = 3x–4; [1, )
43)
A)
0.30
B)
0.32
C)
0.50
D)
0.26
44)
f(x) = 2(1 – x); [0, 1]
44)
A)
0.33
B)
0.29
C)
0.50
D)
0.44
Find the indicated probability.
45)
f(x) = e–x; [0, ), P(x 3)
45)
A)
00.9502
B)
0.0498
C)
0
D)
The function f(x) is not a probability density function.
Solve the problem.
46)
The annual rainfall in Maine is a random variable with probability density function defined by
f(x) =1
15 x +1
2 for x in [0, 5]. Find the standard deviation.
46)
A)
1.270
B)
1.265
C)
1.597
D)
0.903
B
Find the proportion of observations of a standard normal distribution that are between the given z–scores.
47)
0.11 and 2.11
47)
A)
0.4388
B)
0.9388
C)
0.4383
D)
0.4423
A
Find the cumulative distribution function for the given probability density function.
48)
f(x) =3
52 x1/2; [1, 9]
48)
A)
F(x) =x3/2 – 1
26 , 1
x 9
B)
F(x) =x3/2
26 , 1
x 9
C)
F(x) =x1/2 – 1
26 , 1
x 9
D)
F(x) =x3/2 + 1
26 , 1
x 9
A
B
Find the indicated probability.
49)
f(x) =1
4e–x/4; [0, ), P(1 x 6)
49)
A)
0.0695
B)
0.5557
C)
0.0926
D)
0.1389
Find the median of the random variable for the given probability density function.
50)
f(x) =6x–7; [1, )
50)
A)
1.12
B)
0.90
C)
1.07
D)
3.32
Find the indicated probability.
51)
f(x) =
x3
12 if 0 x 2
16
3x3 if x > 2 , P(0 x 2)
51)
A)
1
4
B)
7
24
C)
17
48
D)
1
3
Find the mean and standard deviation of the specified probability density function.
52)
f(x) = 0.07e–0.07x for [0, )
52)
A)
µ= 14.29, = 15.1
B)
µ= 14.1, = 14.29
C)
µ= 14.29, = 14.29
D)
µ= 1.429, = 1.249
Use the standard normal curve table to find the z–score for the given condition.
53)
3.01% of the total area is to the right of z.
53)
A)
1.88
B)
–1.88
C)
1.89
D)
–1.89
Find a value of k that will make f a probability density function on the indicated interval.
54)
f(x) = kx; [2, 4]
54)
A)
1
12
B)
1
16
C)
1
6
D)
1
8
Find the expected value of the probability density function to the nearest hundredth.
55)
f(x) =x
8–1
4; [2, 6]
55)
A)
4.00
B)
5.00
C)
4.33
D)
4.67
Find the median of the random variable for the given probability density function.
56)
f(x) =1
8; [5, 13]
56)
A)
9
B)
23
3
C)
11
D)
5
Find the proportion of observations of a standard normal distribution that are between the given z–scores.
57)
0 and 0.35
57)
A)
0.6368
B)
0.1406
C)
0.3632
D)
0.1368
Find the mean and standard deviation of the specified probability density function.
58)
f(x) = 0.04e–0.04x for [0, )
58)
A)
µ= 25, = 25
B)
µ= 25, = 15
C)
µ= 2.5, = 2.5
D)
µ= 20, = 20
Decide whether or not the function is a probability density function on the indicated interval.
59)
f(x) =4
81 x3; [0, 3]
59)
A)
No
B)
Yes
Find the probability to the nearest hundredth that the random variable of the probability density function has a value
greater than the mean.
60)
f(x) = 4x–5; [1, )
60)
A)
0.32
B)
0.29
C)
0.23
D)
0.50
61)
f(x) =x
3–1
6; [3, 4]
61)
A)
0.54
B)
0.49
C)
0.51
D)
0.57
Find the indicated probability.
62)
f(x) =1
2(1 + x)–3/2; [0, ), P(x 24)
62)
A)
–1
24
B)
1
5
C)
–1
5
D)
1
24
Find the mean and standard deviation of the specified probability density function.
63)
f(x) = e–x for [0, )
63)
A)
µ= 0.5, = 1
B)
µ= 0.5, = 0.5
C)
µ= 10, = 10
D)
µ= 1, = 1
Find the standard deviation of the probability density function to the nearest hundredth.
64)
f(x) =x
8–1
4; [2, 6]
64)
A)
0.92
B)
0.93
C)
0.94
D)
0.95
Decide whether or not the function is a probability density function on the indicated interval.
65)
f(x) =1
9x –1
18 ; [2, 5]
65)
A)
No
B)
Yes
Find the probability to the nearest hundredth that the random variable of the probability density function has a value
greater than the mean.
66)
f(x) =1
3; [3, 6]
66)
A)
0.48
B)
0.53
C)
0.41
D)
0.50
Find the proportion of observations of a standard normal distribution that are between the given z–scores.
67)
–3.0 and 2.65
67)
A)
0.4956
B)
0.0037
C)
0.9946
D)
0.9961
Find the indicated probability.
68)
f(x) =6
(x +6)2; [0, ), P(1 x 6)
68)
A)
0.2381
B)
0.3571
C)
0.5000
D)
The function f(x) is not a probability density function.
Find the probability to the nearest hundredth that the value of the random variable is within one standard deviation of
the mean.
69)
f(x) =1
4; [0, 4]
69)
A)
0.58
B)
0.67
C)
0.75
D)
0.50
Find the standard deviation of the probability density function to the nearest hundredth.
70)
f(x) = 4x–5; [1, ]
70)
A)
0.44
B)
0.50
C)
0.47
D)
0.57
Use the standard normal curve table to find the z–score for the given condition.
71)
4.01% of the total area is to the right of z.
71)
A)
–1.75
B)
1.74
C)
1.75
D)
1.76
72)
20.05% of the total area is to the right of z.
72)
A)
0.82
B)
–0.84
C)
0.83
D)
0.84
Find the variance of the probability density function to the nearest hundredth.
73)
f(x) =1
5; [0, 5]
73)
A)
0
B)
1.04
C)
2.09
D)
2.08
Solve the problem.
74)
The time to failure t, in hours, of a certain machine can often be assumed to be exponentially
distributed with probability density function
f(t) =1
81 e–t/81, 0 t <,
What is the probability that a failure will occur in 50 hours or less?
74)
A)
0.3454
B)
0.4606
C)
0.3685
D)
0.5527
Find the proportion of observations of a standard normal distribution that are between the given z–scores.
75)
–2.34 and –1.1
75)
A)
0.1263
B)
0.1260
C)
0.1249
D)
0.6261
Find the cumulative distribution function for the given probability density function.
76)
f(x) =1
5x –1
10 ; [2, 4]
76)
A)
F(x) =x2– x – 2
10 , 2 x
4
B)
F(x) =x2– x + 2
10 , 2 x
4
C)
F(x) =x2– x – 2
5, 2 x
4
D)
F(x) =x2– x – 4
10 , 2 x
4
Find the standard deviation of the probability density function to the nearest hundredth.
77)
f(x) = 3x–4; [1, ]
77)
A)
0.85
B)
0.87
C)
0.83
D)
0.71