Exhibit 7-5
Random samples of size 17 are taken from a population that has 200 elements, a mean of 36, and a
standard deviation of 8.
89. Refer to Exhibit 7-5. The mean and the standard deviation of the sampling distribution of the sample
means are
a.
8.7 and 1.94
b.
36 and 1.94
c.
36 and 1.86
d.
36 and 8
90. Refer to Exhibit 7-5. Which of the following best describes the form of the sampling distribution of the
sample mean for this situation?
a.
Approximately normal because the sample size is small relative to the population size.
b.
Approximately normal because of the central limit theorem.
c.
exactly normal
d.
None of the alternative answers is correct.
91. The standard error of the proportion will become larger as
a.
n increases
b.
p approaches 0
c.
p approaches .5
d.
p approaches 1
92. All of the following are true about the standard error of the mean except
a.
it is larger than the standard deviation of the population
b.
it decreases as the sample size increases
c.
its value is influenced by the standard deviation of the population
d.
it measures the variability in sample means
PROBLEM
1. A simple random sample of 8 employees of a corporation provided the following information.
Employee
1
2
3
4
5
6
7
8
Age
25
32
26
40
50
54
22
23
Gender
M
M
M
M
F
M
M
F
a.
b.
c.
a.
34
b.
12.57
2. Starting salaries of a sample of five management majors along with their genders are shown below.
Employee
Salary ($1000s)
Gender
1
30
F
2
28
M
3
22
F
4
26
F
5
19
M
a.
b.
c.
a.
25 (thousands of dollars)
b.
20 (thousands of dollars)
c.
0.4
3. A sample of 8 new models of automobiles provides the following data on highway miles per gallon.
Use Excel to answer the questions that follow the data.
Highway Miles
Model
Per Gallon
1
33.6
2
26.8
3
20.2
4
38.7
5
35.1
6
28.0
7
26.2
8
27.6
a.
b.
c.
0.25
4. A sample of 10 members of a video club provides the following data on number of videos they own.
Use Excel to answer the questions that follow the data.
Member
Number Owned
1
200
2
26
3
158
4
75
5
52
6
352
7
17
8
276
9
488
10
129
a.
b.
5. Consider a population of five weights identical in appearance but weighing 1, 3, 5, 7, and 9 ounces.
a.
Determine the mean and the variance of the population.
b.
Sampling without replacement from the above population with a sample size of 2 produces ten
possible samples. Using the ten sample mean values, determine the mean of the population and
the variance of .
c.
Compute the standard error of the mean.
ANS:
6. Consider a population of five families with the following data representing the number of pets in each
family.
Family
Number of Pets
A
2
B
6
C
4
D
3
E
1
a.
b.
c.
a.
b.
c.
7. The following information gives the number of days absent from work for a population of 5 workers at
a small factory.
Number of
Worker
Days Absent
A
5
B
7
C
1
D
4
E
8
a.
b.
a.
5 and 8
b.
5 and 3
c.
1.732
c.
d.
e.
8. The average weekly earnings of bus drivers in a city are $950 (that is
) with a standard deviation of
$45 (that is
). Assume that we select a random sample of 81 bus drivers.
a.
Assume the number of bus drivers in the city is large compared to the sample size. Compute
the standard error of the mean.
b.
What is the probability that the sample mean will be greater than $960?
c.
If the population of bus drivers consisted of 400 drivers, what would be the standard error of
the mean?
5
b.
0.0228
4.47
9. An automotive repair shop has determined that the average service time on an automobile is 2 hours
with a standard deviation of 32 minutes. A random sample of 64 services is selected.
a.
What is the probability that the sample of 64 will have a mean service time greater than 114
minutes?
b.
Assume the population consists of 400 services. Determine the standard error of the mean.
0.9332
b.
3.67
10. A population of 1,000 students spends an average of $10.50 a day on dinner. The standard deviation of
the expenditure is $3. A simple random sample of 64 students is taken.
a.
What are the expected value, standard deviation, and shape of the sampling distribution of the
sample mean?
b.
What is the probability that these 64 students will spend a combined total of more than
$715.21?
c.
What is the probability that these 64 students will spend a combined total between $703.59 and
$728.45?
10.5; 0.363; normal
5; 2.449
b.
5; 1.5
AB, AC, AD, AE, BC, BD, BE, CD, CE, DE
d.
6, 3, 4.5, 6.5, 4, 5.5, 7.5, 2.5, 4.5, 6
11. There are 8,000 students at the University of Tennessee at Chattanooga. The average age of all the
students is 24 years with a standard deviation of 9 years. A random sample of 36 students is selected.
a.
Determine the standard error of the mean.
b.
What is the probability that the sample mean will be larger than 19.5?
c.
What is the probability that the sample mean will be between 25.5 and 27 years?
a.
1.5
b.
0.9986
c.
0.1359
12. The life expectancy in the United States is 75 with a standard deviation of 7 years. A random sample
of 49 individuals is selected.
a.
What is the probability that the sample mean will be larger than 77 years?
b.
What is the probability that the sample mean will be less than 72.7 years?
c.
What is the probability that the sample mean will be between 73.5 and 76 years?
d.
What is the probability that the sample mean will be between 72 and 74 years?
e.
What is the probability that the sample mean will be larger than 73.46 years?
a.
0.0228
b.
0.0107
d.
0.1573
e.
0.9382
13. Scores for a standardized test are normally distributed with a mean of 1200 and a standard deviation of
60. A sample of 36 scores is selected.
a.
What is the probability that the sample mean will be larger than 1224?
b.
What is the probability that the sample mean will be less than 1230?
c.
What is the probability that the sample mean will be between 1200 and 1214?
d.
What is the probability that the sample mean will be greater than 1200?
e.
What is the probability that the sample mean will be larger than 73.46?
a.
0.0082
b.
0.9986
c.
0.4192
d.
0.5
e.
1.0
b.
0.0314
c.
0.0794
14. A bank has kept records of the checking balances of its customers and determined that the average
daily balance of its customers is $300 with a standard deviation of $48. A random sample of 144
checking accounts is selected.
a.
What is the probability that the sample mean will be more than $306.60?
b.
What is the probability that the sample mean will be less than $308?
c.
What is the probability that the sample mean will be between $302 and $308?
d.
What is the probability that the sample mean will be at least $296?
15. Students of a large university spend an average of $5 a day on lunch. The standard deviation of the
expenditure is $3. A simple random sample of 36 students is taken.
a.
What are the expected value, standard deviation, and shape of the sampling distribution of the
sample mean?
b.
What is the probability that the sample mean will be at least $4?
c.
What is the probability that the sample mean will be at least $5.90?
a.
5.0; 0.5; normal
b.
0.9772
c.
0.0359
16. The average lifetime of a light bulb is 3,000 hours with a standard deviation of 696 hours. A simple
random sample of 36 bulbs is taken.
a.
What are the expected value, standard deviation, and shape of the sampling distribution of ?
b.
What is the random variable in this problem? Define it in words.
c.
What is the probability that the average life in the sample will be between 2,670.56 and
2,809.76 hours?
d.
What is the probability that the average life in the sample will be greater than 3,219.24 hours?
e.
What is the probability that the average life in the sample will be less than 3,180.96 hours?
a.
3,000; 116; normal
b.
d.
0.0294
e.
0.9406
17. MNM Corporation gives each of its employees an aptitude test. The scores on the test are normally
distributed with a mean of 75 and a standard deviation of 15. A simple random sample of 25 is taken
from a population of 500.
a.
What are the expected value, the standard deviation, and the shape of the sampling distribution
of ?
a.
0.0495
b.
0.9772
c.
0.2857
d.
0.8413
b.
What is the random variable in this problem? Define it in words.
c.
What is the probability that the average aptitude test score in the sample will be between 70.14
and 82.14?
d.
What is the probability that the average aptitude test score in the sample will be greater than
82.68?
e.
What is the probability that the average aptitude test score in the sample will be less than
78.69?
f.
Find a value, C, such that P( ) = .015.
18. The price of a particular brand of jeans has a mean of $37.99 and a standard deviation of $7. A sample
of 49 pairs of jeans is selected. Use Excel to answer the following questions.
a.
What is the probability that the sample of jeans will have a mean price less than $40?
b.
What is the probability that the sample of jeans will have a mean price between $38 and $39?
c.
What is the probability that the sample of jeans will have a mean price within $3 of the
population mean?
a.
c.
19. The mean diameter of a ball bearing produced by a certain manufacturer is 0.80 cm with a standard
deviation of 0.03 cm. A sample of 36 ball bearings is randomly selected from a production run. Use
Excel to answer the following questions.
a.
What is the probability that the sample of ball bearings will have a mean less than 0.798 cm?
b.
What is the probability that the sample of ball bearings will have a mean of at least 0.815 cm?
c.
What is the probability that the sample of ball bearings will have a mean between 0.798 and
0.815 cm?
d.
For samples of size 36, 15% of all sample means are at most what diameter?
a.
b.
c.
a.
75; 3; normal
b.
c.
0.9387
d.
0.0052
e.
0.8907
81.51
20. There are 500 employees in a firm, 45% are female. A sample of 60 employees is selected randomly.
a.
Determine the standard error of the proportion.
b.
What is the probability that the sample proportion of females is between 0.40 and 0.55?
a.
0.0603
b.
0.7482
21. Ten percent of the items produced by a machine are defective. A random sample of 100 items is
selected and checked for defects.
a.
Determine the standard error of the proportion.
b.
What is the probability that the sample will contain more than 2.5% defective units?
c.
What is the probability that the sample will contain more than 13% defective units?
a.
0.03
b.
0.9938
c.
0.1587
22. A new soft drink is being market tested. It is estimated that 60% of consumers will like the new drink.
A sample of 96 taste-tested the new drink.
a.
Determine the standard error of the proportion
b.
What is the probability that more than 70.4% of consumers will indicate they like the drink?
c.
What is the probability that more than 30% of consumers will indicate they do not like the
drink?
a.
0.05
b.
0.0188
c.
0.9772
23. In a large university, 20% of the students are business majors. A random sample of 100 students is
selected, and their majors are recorded.
a.
Compute the standard error of the proportion.
b.
What is the probability that the sample contains at least 12 business majors?
c.
What is the probability that the sample contains less than 15 business majors?
d.
What is the probability that the sample contains between 12 and 14 business majors?
a.
0.04
b.
0.9772
d.
.794818
=NORM.INV(0.15,0.8,0.005)
24. In a local university, 10% of the students live in the dormitories. A random sample of 100 students is
selected for a particular study.
a.
What is the probability that the sample proportion of students living in the dormitories is
between 0.172 and 0.178?
b.
What is the probability that the sample proportion of students living in the dormitories is
greater than 0.025?
0.0035
25. A department store has determined that 25% of all their sales are credit sales. A random sample of 75
sales is selected.
a
b.
c.
d.
b.
0.8411
0.5
26. Candidate A is running for president of the student government at a large university. The proportion of
voters who favor the candidate is 0.8. A simple random sample of 100 voters is taken.
a.
What are the expected value, standard deviation, and shape of the sampling distribution of ?
b.
What is the probability that the number of voters in the sample who will not favor Candidate A
will be between 26 and 30?
c.
What is the probability that the number of voters in the sample who will not favor Candidate A
will be more than 16?
0.8; 0.04; normal
0.8413
27. In a restaurant, the proportion of people who order coffee with their dinner is 0.9. A simple random
sample of 144 patrons of the restaurant is taken.
a.
What are the expected value, standard deviation, and shape of the sampling distribution of ?
b.
What is the random variable in this problem? Define it in words.
c.
What is the probability that the proportion of people who will order coffee with their meal is
between 0.85 and 0.875?
d.
0.044
d.
What is the probability that the proportion of people who will order coffee with their meal is at
least 0.945?
28. Thirty percent of a magazine’s subscribers are female. A random sample of 50 subscribers is taken.
Answer the following questions using Excel.
a.
What is the probability that the proportion of females from this sample is at most 0.25?
b.
What is the probability that the proportion of females from this sample is between 0.22 and
0.28?
c.
What is the probability that the proportion of females from this sample is within .03 of the
population proportion?
b.
29. The proportion of Americans who support the death penalty is 0.53. A sample of 1000 randomly
selected Americans is surveyed by telephone interview. Use Excel to answer the following questions.
a.
What is the probability that the sample proportion of those supporting the death penalty will be
less than 0.50?
b.
What is the probability that the sample proportion of those supporting the death penalty will be
at least 0.55?
c.
What is the probability that the sample proportion of those supporting the death penalty will be
between 0.50 and 0.55?
d.
For samples of size 1000, 15% of all sample proportions are at most what value?
a.
d.
30. A random sample of nine telephone calls in an office provided the following information.
a.
0.9; 0.025; normal
c.
0.1359
d.
0.0359
Call Number
Duration (Minutes)
Type of Call
1
3
local
2
8
long distance
3
4
local
4
3
local
5
5
long distance
6
6
local
7
3
local
8
5
local
9
8
local
a.
b.
c.
31. A random sample of ten examination papers in a course that was given on a pass or fail basis showed
the following scores.
Paper Number
Grade
Status
1
65
Pass
2
87
Pass
3
92
Pass
4
35
Fail
5
79
Pass
6
100
Pass
7
48
Fail
8
74
Pass
9
79
Pass
10
91
Pass
a.
b.
c.
a.
75
b.
20.48
c.
0.8
32. Roger Hall, who oversees six Ford dealerships, believes that the colors chosen by customers who
special-order their cars best reflect most customers’ true color preferences. For that reason, he has
tabulated the color requests specified in a sample of 56 Mustang coupe special orders placed this year.
The sample data are listed below.
b.
1.97
c.
0.222
Black
Red
White
Blue
Blue
Green
Red
Black
Red
White
Blue
White
Red
Red
Black
Black
Green
Black
Red
Black
Blue
Black
White
Green
Blue
Red
Black
White
Black
Red
Black
Blue
Blue
Black
Green
White
Black
Red
Red
White
Red
Red
Blue
Black
Red
Black
Green
Black
Green
Red
Black
White
Black
Red
Black
White
a. What is the point estimate of the proportion of all Mustang coupe special orders that specify a
color preference of black?
b. Describe the sampling distribution of , where is the proportion of Mustang coupe special
orders that specify a color preference of black. Assume that the proportion of all Mustang coupe
special orders having a color preference of black is .36.
c. What is the probability that a simple random sample of 56 special orders will provide an estimate
of the population proportion of special orders specifying the color black that is within plus or
minus .05 of the actual population proportion, assuming p = .36? In other words, what is the
probability that will be between .31 and .41?
33. Missy Walters owns a mail-order business specializing in baby clothes. Missy is confident the dollar
amounts of all her orders are normally distributed or nearly so. Assume she knows the mean and
standard deviation are $249 and $46, respectively, for all orders she receives.
a. Describe the sampling distribution of , where is the mean dollar-amount of an order for a
sample of 10 orders.
b. What is the probability that a simple random sample of 30 orders will provide an estimate of the
population mean dollar-amount of an order that is within plus or minus $10 of the actual
population mean?
c. What happens to the sampling distribution of when the sample size is increased from 30 to 90?
With a sample size of 90, what is the probability that will be between $239 and $259?