Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1)
The life (in hours) of light bulbs of a certain brand is normally distributed with mean 1200 and
standard deviation 100. What percentage of such bulbs will burn more than 1250 hours?
1)
A)
20.47
B)
13.62
C)
30.85
D)
48.26
E)
41.73
2)
If X has density function f(x) =x+1
2,if 0 x
1,
0, otherwise,
find µ.
2)
A)
1
5
B)
1
6
C)
5
6
D)
7
12
E)
5
3
3)
If X has density function f(x) =kx2,if 0 x1
2,
0, otherwise,
find k.
3)
A)
1
24
B)
12
C)
2
D)
24
E)
23
24
4)
Suppose X is normally distributed with mean 100 and standard deviation 20. Without using tables,
determine the approximate value of P(80 <X< 100).
4)
A)
0.68
B)
0.34
C)
0.997
D)
0.50
E)
0.95
1
5)
Suppose X is uniformly distributed over the interval 2, 10 . Find P(3 <X< 8).
5)
A)
1
2
B)
3
8
C)
5
8
D)
2
5
E)
3
5
6)
If X has density function f(x) =2x
9,if 0 x 3,
0, otherwise,
find µ.
6)
A)
2
27
B)
1
9
C)
2
9
D)
1
2
E)
2
7)
If X is normally distributed with µ= 50 and = 10, find x0 such that P(X<x0) = 0.9938.
7)
A)
100
B)
75
C)
80
D)
25
E)
50
8)
The life (in hours) of light bulbs of a certain brand is normally distributed with mean 1000 and
standard deviation 100. What percentage of such bulbs will burn more than 950 hours?
8)
A)
40.95
B)
69.15
C)
68.34
D)
65.54
E)
72.15
2
9)
Suppose X is normally distributed with mean 80 and standard deviation 15. Without using tables,
determine the approximate value of P(65 <X< 95).
9)
A)
0.997
B)
0.33
C)
0.50
D)
0.95
E)
0.68
10)
Suppose X has density function f(x) =x
4,if 1 x
3,
0, otherwise.
Find P(2 <X< 3).
10)
A)
1
2
B)
1
3
C)
3
8
D)
5
8
E)
1
8
11)
Suppose X is uniformly distributed over the interval 2, 7 . Find P(X> 5).
11)
A)
03.
B)
0.4
C)
0.5
D)
0.6
E)
0.7
12)
Suppose X has a binomial distribution with n= 100 and p= 0.1. Using the normal approximation,
find P(X
7).
12)
A)
0.4126
B)
0.3128
C)
0.2514
D)
0.3467
E)
0.2033
13)
If X is normally distributed with µ= 20 and = 5, find x0 such that P(X>x0) = 0.025.
13)
A)
29.8
B)
18.6
C)
21.96
D)
31.6
E)
27.2
14)
If X has density function f(x) =x
8,if 0 x
4,
0, otherwise,
find c such that P(X<c) =1
4.
14)
A)
2 2
B)
8
C)
4 2
D)
4
3
E)
2
15)
If X is normally distributed with mean µ and standard deviation , then what is the approximate
probability that µ– 2Xµ+ 2?
15)
A)
0.95
B)
0.68
C)
0.997
D)
0.5
E)
0.999
16)
If a fair coin is tossed 100 times, use the normal approximation to estimate the probability of getting
at least 60 heads.
16)
A)
0.0179
B)
0.4713
C)
0.5287
D)
0.0228
E)
0.0287
17)
Suppose X has density function f(x) =2e–2x,if x
0,
0, if x< 0. Find P(X< 1).
17)
A)
2e–2– 2
B)
e–2– 1
C)
e–2
D)
1 – 2e–2
E)
1 –e–2
18)
If X is normally distributed with µ= 80 and = 8, find P(X< 74).
18)
A)
0.7266
B)
0.2266
C)
0.2734
D)
0.7734
E)
0.7732
19)
If X is normally distributed with µ= 150 and = 30, find P(90 <X< 180).
19)
A)
0.8243
B)
0.1359
C)
0.3185
D)
0.8185
E)
0.8173
20)
If X is normally distributed with µ= 60 and = 5, find P(X> 72).
20)
A)
0.2731
B)
0.0123
C)
0.2968
D)
0.0082
E)
0.4918
21)
If X has density function f(x) =x+1
2,if 0 x
1,
0, otherwise,
find .
21)
A)
11
12
B)
5
12
C)
35
12
D)
7
12
E)
1
2
22)
If X has density function f(x) =x
3k,if 0 x 1,
0, otherwise,
find k.
22)
A)
1
2
B)
6
C)
0
D)
1
6
E)
1
5
23)
If X is normally distributed with µ= 90 and = 10, find P(65 <X< 80).
23)
A)
0.1428
B)
0.1525
C)
0.1232
D)
0.1048
E)
0.1347
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
24)
The random variable X has density function f(x) =5e–5x,if x
0,
0, if x< 0.
Find P(X 4). Give answer in terms of e.
24)
25)
The life expectancy (in years) of patients after they have contracted a certain disease is
exponentially distributed with k= 0.4. Find the mean life expectancy and the standard
deviation.
25)
26)
Suppose the cumulative distribution function for the random variable X is given by
F(x) =
0, if x< 0,
x2
25 ,if 0 x 5,
1, if x> 5.
Find P(2 <X< 3).
26)
27)
If Z is a standard normal variable, find z0 such that P(–z0<Z<z0 ) = 0.823.
27)
28)
Playing a fair “shell game,” the probability of winning the game should be given by p=1
3.
One person plays the game 50 times but only wins 7 times. Using n= 50, approximate P(X
= 7) by using the normal approximation.
28)
29)
If X is uniformly distributed over 3, 6 , what is the density function for X?
29)
30)
The life expectancy (in years) of a keyboard is distributed exponentially with k=1
5. If the
keyboard’s warranty lasts 2 years, what is the probability that a keyboard will break down
after the warranty expires?
30)
31)
The random variable X has density function f(x) =4e–4x,if x
0,
0, if x< 0.
Find P(X> 2). Give answer in terms of e.
31)
32)
The random variable X has density function f(x) =6x(1 –x), if 0 x
1,
0, otherwise.
Find (a) P1
3< X <2
3 and (b) P X >2
3.
32)
33)
If X is a normal random variable with µ= 16 and = 2, determine the Z–value that
corresponds to X= 23.
33)
34)
The scores on an examination are normally distributed with µ= 500 and = 100. What
percentage of the scores are less than 360?
34)
35)
The scores on an examination are normally distributed with µ= 75 and = 10. What
percentage of the scores are between 70 and 90?
35)
36)
The random variable X has density function f(x) =2x,if 0 x
1,
0, otherwise.
Find the cumulative distribution function F over 0, 1 .
36)
37)
If Z has a standard normal distribution, find P(Z> –1.81).
37)
7
38)
The random variable X has density function f(x) =x
8,if 0 x
4,
0, otherwise.
Find µ and .
38)
39)
If X is a normal random variable with µ= 30 and = 5, determine the Z–value that
corresponds to X= 28.
39)
40)
Suppose the time (in minutes) applicants must wait to receive a driver’s examination is
uniformly distributed with density function f(x) =1
40 where 0 x
40 and f(x) = 0
elsewhere. What is the probability that an applicant must wait more than 35 minutes?
40)
41)
If Z has a standard normal distribution, find P(Z> 2).
41)
42)
If X is normally distributed with µ= 18 and = 2, find P(11
X
16).
42)
43)
If X is normally distributed with µ= 8 and = 2, find P(3 X
8).
43)
44)
Suppose the time (in minutes) passengers must wait for an airplane is uniformly
distributed with density function f(x) =1
60 where 0 x
60 and f(x) = 0 elsewhere. What is
the probability that a commuter must wait more than 0 minutes?
44)
45)
Suppose X is normally distributed with µ= 80 and = 15. Without using tables,
approximate P(50 <X< 80).
45)
8
46)
Suppose X is uniformly distributed over the interval 0, 4 . Find (a) P(2 X
3) and (b) P(X
2).
46)
47)
The random variable X has density function f(x) =x+1
2,if 0 x
1,
0, otherwise.
Find the cumulative distribution function F over 0, 1 .
47)
48)
If Z has a standard normal distribution, find P(Z< –1.23).
48)
49)
If Z has a standard normal random variable and P(Z>z0) = 0.4539, find P(0 <Z<z0).
49)
50)
If X is normally distributed with µ= 50 and = 8, find P(X 44).
50)
51)
If Z has a standard normal distribution, state the mean and standard deviation of Z.
51)
52)
If X is normally distributed with µ= 25 and = 5, find P(20
X
35).
52)
53)
The life expectancy (in years) of a computer printer is distributed exponentially with k=1
4.
If the printer’s warranty lasts 3 years, what is the probability that a printer will last more
than 10 years?
53)
54)
If X is normally distributed with µ= 100 and = 10, find P(X< 115).
54)
9
55)
The length of time (in minutes) that a person arriving at a train station must wait for a train
is uniformly distributed with density function f(x) =1
20 where 0 x
20. Find the mean
waiting time and the standard deviation.
55)
56)
The random variable X has density function f(x) =3x2,if 0 x 1,
0, otherwise.
Find µ and .
56)
57)
If X is uniformly distributed over 2, 8 , what is the density function for X?
57)
58)
Suppose X has binomial distribution with n= 100 and p= 0.1. Using the normal
approximation, find (a) P(X= 13) and (b) P(X
16).
58)
59)
The random variable X has density function f(x) =x
8,if 0 x
4,
0, otherwise.
Find the cumulative distribution function F over 0, 4 .
59)
60)
If Z is a standard normal variable, find z0 such that P( Zz0 ) = 0.9918.
60)
61)
The random variable X has density function f(x) =2x,if 0 x
1
0, otherwise.
Find (a) P1
4 X 1
2 and (b) P X 3
4.
61)
62)
If X is normally distributed with µ= 10 and = 2, find P(4
X 12).
62)
10
63)
If Z has a standard normal distribution, find P(–2 Z–1).
63)
64)
If Z has a standard normal distribution, find P(|Z| <1
4).
64)
65)
If Z has a standard normal distribution, find P(2.2 <Z< 3.5).
65)
66)
A professor claims that there is only a 15% chance of earning an “A” in her class. At the end
of the semester, 40 of the 278 students earn “A’s.” Using n= 278 and p= 0.15, approximate
P(X= 40) by using the normal approximation.
66)
67)
The life expectancy (in years) of a computer printer is distributed exponentially with k=1
4.
If the printer’s warranty lasts 3 years, what is the probability that a printer will break down
during the warranty period?
67)
68)
If X is normally distributed with µ= 12 and = 2, find P(12
X
15).
68)
69)
If Z has a standard normal distribution, find P(2.5 Z 3).
69)
70)
If the density function for the random variable X is f(x) =kx +1
4,if 0 x 1
0, otherwise
,
find k.
70)
71)
If X is normally distributed with µ= 100 and = 10, find P(X< 118).
71)
72)
If Z has a standard normal distribution, find P(–1 Z 2).
72)
73)
Suppose X has binomial distribution with n= 100 and p=9
25. Using the normal
approximation, find (a) P(X= 40) and (b) P(X
26).
73)
74)
If Z has a standard normal distribution, find P(–3 Z–0.5).
74)
75)
A manufacturing plant claims that only 0.2% of its products are defective. From a sample
of 3000 products, it is found that 4 are defective. Using n= 3000 and p= 0.002, approximate
P(X= 4) by using the normal approximation.
75)
76)
If Z has a standard normal distribution, find P(–0.65 <Z< 1.92).
76)
77)
If X is normally distributed with µ= 20 and = 4, find P(16
77)
78)
If Z has a standard normal distribution, find P(0 <Z< 1.48).
78)
79)
If Z has a standard normal distribution, find P(|Z| > 1.75).
79)
80)
If X is normally distributed with µ= 40 and = 6, find P(X 37).
80)
81)
The random variable X has density function f(x) =10
x2,if x
10,
0, if x< 10.
Find (a) P(20 X
40) and (b) P(X 40).
81)
82)
The length of time (in minutes) that a person arriving at a bus stop must wait for a bus is
uniformly distributed with density function f(x) =1
15 where 0 x
15. Find the mean
waiting time and the standard deviation.
82)
83)
If Z has a standard normal distribution, find P(Z> 0.12).
83)
84)
Suppose the cumulative distribution function for the random variable X is given by
F(x) =
0, if x< 0,
x
4,if 0 x
4,
1, if x> 4.
Find P(1 <X< 3).
84)
85)
If Z has a standard normal random variable and P(Z>z0) = 0.9938, find P(z0<Z< 0).
85)
86)
Suppose X is uniformly distributed over the interval 1, 6 . Find P(2 <X< 4).
86)
87)
If the density function for the random variable X is f(x) =kx2,if 0 x 3
0, otherwise , find k.
87)
88)
Suppose the time (in minutes) applicants must wait to receive a driver’s examination is
uniformly distributed with density function f(x) =1
40 where 0 x
40 and f(x) = 0
elsewhere. What is the probability that an applicant must wait less than 25 minutes?
88)
13
Answer Key
Testname: C16
14
Answer Key
15
Answer Key
Testname: C16