Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1)
If X is normally distributed with mean µ and standard deviation , then what is the approximate
probability that µ– 2Xµ+ 2?
1)
A)
0.997
B)
0.5
C)
0.999
D)
0.95
E)
0.68
2)
If X is normally distributed with µ= 80 and = 8, find P(X< 74).
2)
A)
0.2266
B)
0.7266
C)
0.2734
D)
0.7732
E)
0.7734
3)
If X has density function f(x) =x+1
2,if 0 x
1,
0, otherwise,
find µ.
3)
A)
1
5
B)
1
6
C)
5
6
D)
7
12
E)
5
3
4)
If X has density function f(x) =x+1
2,if 0 x
1,
0, otherwise,
find .
4)
A)
11
12
B)
5
12
C)
35
12
D)
7
12
E)
1
2
1
5)
Suppose X has a binomial distribution with n= 100 and p= 0.1. Using the normal approximation,
find P(X
7).
5)
A)
0.2514
B)
0.3467
C)
0.2033
D)
0.3128
E)
0.4126
6)
If a fair coin is tossed 100 times, use the normal approximation to estimate the probability of getting
at least 60 heads.
6)
A)
0.0287
B)
0.0228
C)
0.5287
D)
0.4713
E)
0.0179
7)
If X has density function f(x) =2x
9,if 0 x 3,
0, otherwise,
find µ.
7)
A)
2
27
B)
1
9
C)
2
9
D)
1
2
E)
2
8)
Suppose X is normally distributed with mean 100 and standard deviation 20. Without using tables,
determine the approximate value of P(80 <X< 100).
8)
A)
0.50
B)
0.997
C)
0.68
D)
0.34
E)
0.95
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)
0.50
B)
0.33
C)
0.95
D)
0.997
E)
0.68
10)
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?
10)
A)
72.15
B)
40.95
C)
69.15
D)
65.54
E)
68.34
11)
If X is normally distributed with µ= 50 and = 10, find x0 such that P(X<x0) = 0.9938.
11)
A)
25
B)
100
C)
80
D)
75
E)
50
12)
If X has density function f(x) =kx2,if 0 x1
2,
0, otherwise,
find k.
12)
A)
2
B)
12
C)
23
24
D)
24
E)
1
24
13)
If X is normally distributed with µ= 60 and = 5, find P(X> 72).
13)
A)
0.4918
B)
0.0082
C)
0.2968
D)
0.2731
E)
0.0123
14)
Suppose X has density function f(x) =x
4,if 1 x
3,
0, otherwise.
Find P(2 <X< 3).
14)
A)
1
2
B)
1
3
C)
3
8
D)
5
8
E)
1
8
15)
Suppose X has density function f(x) =2e–2x,if x
0,
0, if x< 0. Find P(X< 1).
15)
A)
2e–2– 2
B)
e–2– 1
C)
e–2
D)
1 – 2e–2
E)
1 –e–2
16)
If X has density function f(x) =x
8,if 0 x
4,
0, otherwise,
find c such that P(X<c) =1
4.
16)
A)
4
3
B)
2
C)
4 2
D)
2 2
E)
8
17)
If X is normally distributed with µ= 90 and = 10, find P(65 <X< 80).
17)
A)
0.1232
B)
0.1428
C)
0.1048
D)
0.1347
E)
0.1525
18)
If X has density function f(x) =x
3k,if 0 x 1,
0, otherwise,
find k.
18)
A)
6
B)
1
2
C)
1
6
D)
0
E)
1
19)
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?
19)
A)
30.85
B)
20.47
C)
48.26
D)
13.62
E)
41.73
20)
Suppose X is uniformly distributed over the interval 2, 10 . Find P(3 <X< 8).
20)
A)
1
2
B)
3
8
C)
5
8
D)
2
5
E)
3
5
21)
Suppose X is uniformly distributed over the interval 2, 7 . Find P(X> 5).
21)
A)
03.
B)
0.4
C)
0.5
D)
0.6
E)
0.7
22)
If X is normally distributed with µ= 150 and = 30, find P(90 <X< 180).
22)
A)
0.3185
B)
0.8173
C)
0.8185
D)
0.1359
E)
0.8243
23)
If X is normally distributed with µ= 20 and = 5, find x0 such that P(X>x0) = 0.025.
23)
A)
31.6
B)
18.6
C)
21.96
D)
27.2
E)
29.8
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) =x
8,if 0 x
4,
0, otherwise.
Find the cumulative distribution function F over 0, 4 .
24)
25)
If Z has a standard normal distribution, state the mean and standard deviation of Z.
25)
26)
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?
26)
27)
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.
27)
28)
If Z has a standard normal distribution, find P(Z> 2).
28)
29)
If Z is a standard normal variable, find z0 such that P(–z0<Z<z0 ) = 0.823.
29)
30)
The scores on an examination are normally distributed with µ= 500 and = 100. What
percentage of the scores are less than 360?
30)
31)
If Z has a standard normal distribution, find P(2.5 Z 3).
31)
32)
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.
32)
33)
If X is normally distributed with µ= 25 and = 5, find P(20
X
35).
33)
34)
If X is normally distributed with µ= 100 and = 10, find P(X< 118).
34)
35)
If X is normally distributed with µ= 12 and = 2, find P(12
X
15).
35)
36)
If Z has a standard normal distribution, find P(–1 Z 2).
36)
37)
If Z has a standard normal distribution, find P(0 <Z< 1.48).
37)
38)
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?
38)
7
39)
Suppose X is uniformly distributed over the interval 0, 4 . Find (a) P(2 X
3) and (b) P(X
2).
39)
40)
If the density function for the random variable X is f(x) =kx2,if 0 x 3
0, otherwise , find k.
40)
41)
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.
41)
42)
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).
42)
43)
The scores on an examination are normally distributed with µ= 75 and = 10. What
percentage of the scores are between 70 and 90?
43)
44)
If Z has a standard normal distribution, find P(|Z| <1
4).
44)
45)
If X is normally distributed with µ= 20 and = 4, find P(16
X
18).
45)
46)
If X is normally distributed with µ= 10 and = 2, find P(4
X 12).
46)
8
47)
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).
47)
48)
Suppose X is normally distributed with µ= 80 and = 15. Without using tables,
approximate P(50 <X< 80).
48)
49)
If Z has a standard normal distribution, find P(Z> –1.81).
49)
50)
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.
50)
51)
If Z has a standard normal distribution, find P(Z< –1.23).
51)
52)
If X is normally distributed with µ= 40 and = 6, find P(X 37).
52)
53)
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 .
53)
54)
If Z has a standard normal distribution, find P(Z> 0.12).
54)
55)
If X is normally distributed with µ= 8 and = 2, find P(3 X
8).
55)
9
56)
Suppose X is uniformly distributed over the interval 1, 6 . Find P(2 <X< 4).
56)
57)
If X is uniformly distributed over 3, 6 , what is the density function for X?
57)
58)
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).
58)
59)
If X is a normal random variable with µ= 16 and = 2, determine the Z–value that
corresponds to X= 23.
59)
60)
If X is uniformly distributed over 2, 8 , what is the density function for X?
60)
61)
If Z has a standard normal distribution, find P(–0.65 <Z< 1.92).
61)
62)
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.
62)
63)
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.
63)
64)
If X is a normal random variable with µ= 30 and = 5, determine the Z–value that
corresponds to X= 28.
64)
65)
If Z has a standard normal distribution, find P(–2 Z–1).
65)
66)
If Z has a standard normal random variable and P(Z>z0) = 0.9938, find P(z0<Z< 0).
66)
67)
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).
67)
68)
The random variable X has density function f(x) =x
8,if 0 x
4,
0, otherwise.
Find µ and .
68)
69)
If Z has a standard normal distribution, find P(2.2 <Z< 3.5).
69)
70)
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?
70)
71)
The random variable X has density function f(x) =3x2,if 0 x 1,
0, otherwise.
Find µ and .
71)
11
72)
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.
72)
73)
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.
73)
74)
If Z is a standard normal variable, find z0 such that P( Zz0 ) = 0.9918.
74)
75)
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.
75)
76)
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).
76)
77)
If the density function for the random variable X is f(x) =kx +1
4,if 0 x 1
0, otherwise
,
find k.
77)
78)
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?
78)
12
79)
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 .
79)
80)
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?
80)
81)
If Z has a standard normal distribution, find P(–3 Z–0.5).
81)
82)
If Z has a standard normal distribution, find P(|Z| > 1.75).
82)
83)
If X is normally distributed with µ= 18 and = 2, find P(11
X
16).
83)
84)
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.
84)
85)
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?
85)
86)
If X is normally distributed with µ= 100 and = 10, find P(X< 115).
86)
87)
If X is normally distributed with µ= 50 and = 8, find P(X 44).
87)
88)
If Z has a standard normal random variable and P(Z>z0) = 0.4539, find P(0 <Z<z0).
88)
Answer Key
Testname: C16
14
Answer Key
Testname: C16
15
Answer Key
Testname: C16