48. In the formula , the
/ 2 refers to:
a.
the probability that the sample mean will not equal the population mean.
b.
the probability that the confidence interval will not contain the population mean.
c.
the level of confidence.
d.
None of these choices.
49. The larger the confidence level, the:
a.
smaller the value of z
/ 2.
b.
wider the confidence interval.
c.
smaller the probability that the confidence interval will contain the population mean.
d.
None of these choices.
50. A 99% confidence interval estimate of the population mean
can be interpreted to mean:
a.
if all possible sample are taken and confidence intervals created, 99% of them would
include the true population mean somewhere within their interval.
b.
we have 99% confidence that we have selected a sample whose interval does include the
population mean.
c.
we estimate that the population mean falls between the lower and upper confidence limits,
and this type of estimator is correct 99% of the time.
d.
All of these choices are true.
51. Suppose a 95% confidence interval for
turns out to be (1,000, 2,100). What does it mean to be 95%
confident?
a.
In repeated sampling, the population parameter would fall in the resulting interval 95% of
the time.
b.
95% of the observations in the entire population fall in the given interval.
c.
95% of the observations in the sample fall in the given interval.
d.
None of these choices.
52. It is desired to estimate the average total compensation of CEOs in the publishing industry. Data were
randomly collected from 18 CEOs and 95% confidence interval was calculated to be ($2,190,000,
$4,720,000). Based on the interval above, do you believe the actual average total compensation of
CEOs in the publishing industry could be $3,000,000?
a.
Yes, and I am sure of that.
b.
Yes, and I am 95% confident of that.
c.
No, and I am sure of that.
d.
No, and I am 95% confident of that.
53. When estimating the population mean using a confidence interval, the sample mean is in the
____________________ of the interval.
54. A confidence interval is a probability statement about the ____________________ mean.
55. The confidence ____________________ is the chance that the interval contains the parameter, over
repeated sampling.
56. In the formula for a confidence interval for
, 1 −
is called the ____________________.
57. The smallest number in a confidence interval is abbreviated by ____________________.
58. It is ____________________ to say that a 95% confidence interval means there is a 95% chance that
the parameter lies in that confidence interval.
59. The population mean is a(n) ____________________ but ____________________ quantity.
60. If you increase the confidence level, the confidence interval becomes ____________________.
61. We do not have control over the population standard deviation when forming a confidence interval,
but we can control the ____________________ and the ____________________.
62. The confidence level that is considered to be the standard amongst statisticians is
____________________%.
63. If the population values are very close to each other, the width of the confidence interval is
____________________ than if the population values are very far apart.
64. Increasing the sample size fourfold decreases the width of the confidence interval by
____________________.
65. A survey of 100 businesses revealed that the mean after-tax profit was $80,000. Assume the
population standard deviation is $15,000:
a.
Determine the 95% confidence interval estimate of the mean after-tax profit for all
businesses.
b.
Explain why you can use the confidence interval formula here, even though the population
is not necessarily normal.
ANS:
a.
LCL = $77,060 and UCL = $82,940.
66. The temperature readings for 20 fall days (degrees Fahrenheit) in Wheeling, West Virginia, are
normally distributed with a mean of 5.5 degrees and a standard deviation of 1.5 degrees. Determine the
90% confidence interval estimate for the fall mean temperature.
67. A sample of 49 measurements of tensile strength for roof hangers are calculated to have a mean of
2.45 and a standard deviation of 0.25. (Units are Newton’s per square meter.)
a.
Determine the 95% confidence interval for mean tensile strength for all hangers.
b.
Interpret this confidence interval.
a.
LCL = 2.38 and UCL = 2.52 (Newton’s per square meter).
68. A random sample of 10 waiters in Weston, West Virginia, revealed the following hourly earnings,
including tips: 19, 18, 15, 16, 18, 17, 16, 18, 20, and 14. (Units are dollars.)
a.
If the hourly earnings are normally distributed with a standard deviation of $4.50, estimate
with 95% confidence the mean hourly earnings for all waitresses in Iowa City.
b.
Interpret your confidence interval.
LCL = $14.31 and UCL = $19.89.
$19.89. I am 95% confident that in repeated sampling, the population parameter would fall
in the resulting interval.
Time Spent Playing Computer Games
Suppose that the amount of time teenagers spend playing computer games per week is normally
distributed with a standard deviation of 1.5 hours. A sample of 100 teenagers is selected at random,
and the sample mean computed as 6.5 hours.
69. {Time Spent Playing Computer Games Narrative} Determine the 95% confidence interval estimate of
the population mean.
70. {Time Spent Playing Computer Games Narrative} Interpret the 95% confidence interval for this
situation.
71. {Time Spent Playing Computer Games Narrative} Determine and interpret the 99% confidence
interval estimate of the population mean.
72. {Time Spent Playing Computer Games Narrative} Determine and interpret the 90% confidence
interval estimate of the population mean.
73. {Time Spent Playing Computer Games Narrative} Determine the 95% confidence interval estimate of
the population mean if the sample size is changed to 300.
74. {Time Spent Playing Computer Games Narrative} Determine the 95% confidence interval estimate of
the population mean if the sample size is changed to 36.
75. {Time Spent Playing Computer Games Narrative} Determine the 95% confidence interval estimate of
the population mean if the population standard deviation is changed to 2.
76. {Time Spent Playing Computer Games Narrative} Determine the 95% confidence interval estimate of
the population mean if the population standard deviation is changed to 1.2.
77. {Time Spent Playing Computer Games Narrative} Determine the 95% confidence interval estimate of
the population mean if the sample mean is changed to 5.0 hours.
78. {Time Spent Playing Computer Games Narrative} Determine the 95% confidence interval estimate of
the population mean if the sample mean is changed to 8.5 hours.
ANS:
79. {Time Spent Playing Computer Games Narrative} What happens to the width of a confidence interval
in each of the following situations?
a.
Confidence level increases
b.
Confidence level decreases
c.
Sample size increases
d.
Sample size decreases
e.
Population standard deviation increases
f.
Population standard deviation decreases
g.
Sample mean increases
h.
Sample mean decreases
a.
Widens
b.
Narrows
Narrows
d.
Widens
Widens
Narrows
g.
No change
h.
No change
80. A random sample of 10 university students was surveyed to determine the amount of time spent
weekly at the library. The times are: 13, 14, 5, 6, 8, 10, 7, 12, 15, and 3. If the times are normally
distributed with a standard deviation of 5.2 hours, estimate with 90% confidence the mean weekly time
at the library by all university students.
81. A financial consultant wanted to determine the mean annual return on Bond funds. A random sample
of 60 returns shows a mean of 12%. If the population standard deviation is assumed to be 4%, estimate
with 95% confidence the mean annual return on all Bond funds.
82. A market researcher is interested in studying the incomes of consumers in a particular region. The
population standard deviation is known to be $1,000. A random sample of 50 individuals resulted in an
average income of $15,000. What is the upper end point in a 99% confidence interval for the average
income?
83. An economist is interested in studying the spending habits of consumers in a particular region. The
population standard deviation is known to be $1,000. A random sample of 50 individuals resulted in an
average expense of $15,000. What is the width of the 99% confidence interval?
84. A precision control engineer is interested in the mean length of tubing being cut automatically by
machine. It is known that the standard deviation in the cutting length is 0.15 feet. A sample of 60 cut
tubes yields a mean length of 12.15 feet. This sample will be used to obtain a 99% confidence interval
for the mean length cut by machine.
a.
What is the z / 2 value to use in obtaining the confidence interval?
b.
Develop the 99% confidence interval for
.
b.
LCL = 12.10 and UCL = 12.20 (feet)
Conference Rooms Rented
A hotel chain wants to estimate the average number of conference rooms rented daily in each month.
The population of conference rooms rented daily is assumed to be normally distributed for each with a
standard deviation of 24 rooms.
85. {Conference Rooms Rented Narrative} During January, a sample of 16 days has a sample mean of 48
rooms. This information is used to calculate an interval estimate for the population mean to be from 40
to 56 rooms. What is the level of confidence of this interval?
86. {Conference Rooms Rented Narrative} During February, a sample of 25 days has a sample mean of 37
rooms. Use this information to calculate a 92% confidence interval for the population mean.
87. To estimate with 95% confidence the mean of a normal population whose standard deviation is
assumed to be 4 and the maximum allowable sampling error is assumed to be 1, requires a random
sample of size 62.
88. The sample size needed to estimate the population mean was found to be 865. If the value of the
population standard deviation was 75, and the maximum allowable error was 5, then the confidence
level used was 95%.
89. The sample size needed to estimate a population mean to within 1 unit with 90% confidence given that
the population standard deviation is 10 is 17.
90. In determining the size n needed to estimate the population mean, n increases as the confidence level
decreases.
91. In determining the sample size n needed to estimate the population mean, n decreases as the desired
width of the confidence interval decreases.
92. In determining the sample size n needed to estimate the population mean, n is higher if the population
standard deviation
is higher.
93. When determining the sample size necessary for estimating the true population mean, which factor is
not of our concern when sampling with replacement?
a.
The allowable or tolerable sampling error.
b.
The level of confidence desired in the estimate.
c.
The population standard deviation.
d.
The population size.
94. Which of the following statements is false regarding the sample size needed to estimate a population
mean?
a.
It is directly proportional to the population variance.
b.
It is directly proportional to the square of z
/ 2.
c.
It is directly proportional to the square of the maximum allowable error B.
d.
None of these choices.
95. To estimate the mean of a normal population whose standard deviation is 6, with a bound on the error
of estimation equal to 1.2 and confidence level 99% requires a sample size of at least:
a.
166
b.
167
c.
13
d.
None of these choices.
96. The sample size needed to estimate a population mean within 2 units with a 95% confidence when the
population standard deviation equals 8 is
a.
62
b.
61
c.
8
d.
None of these choices.
97. The sample size needed to estimate a population mean to within 10 units was found to be 68. If the
population standard deviation was 50, then the confidence level used was:
a.
99%
b.
95%
c.
90%
d.
None of these choices.
98. The sample size needed to estimate a population mean to within 50 units was found to be 97. If the
population standard deviation was 250, then the confidence level used was
a.
90%
b.
95%
c.
99%
d.
None of these choices.
99. The sampling error for a confidence interval is also defined as the ____________________ of
____________________.
100. The error of estimation is the ____________________ between an estimator and the parameter.
101. When determining the required sample size for a confidence interval, you need to know the population
____________________, the confidence ____________________, and the ____________________
on the error of estimation.
102. Because n is an integer and we want the bound on the error of estimation to be no more than a given
amount, any non-integer value found for n must always be rounded ____________________.
103. If the bound on the error of estimation decreases, the sample size ____________________.
104. If the population standard deviation is guesstimated, and it turned out to be smaller than you assumed,
then the sample size you calculated is ____________________ than it needs to be.
105. If the population standard deviation is guesstimated, and it turned out to be larger than you assumed,
then the sample size you calculated is ____________________ than it needs to be.
106. Statisticians can control the ____________________ of a confidence interval by determining the
sample size necessary to produce the desired results.
107. The bound on the error of estimation is the ____________________ amount of sampling error that we
are willing to tolerate.
108. As the bound on the error of estimation decreases, the sample size ____________________.
109. Determine the sample size that is required to estimate a population mean to within 0.4 units with a
99% confidence when the population standard deviation is 1.75.
College Dean
A college dean would like to estimate a population mean to within 40 units with 99% confidence given
that the population standard deviation is 200.
110. {College Dean Narrative} What sample size should be used?
111. {College Dean Narrative} What sample size should be used if the standard deviation is changed to 50?
112. {College Dean Narrative} What sample size should be used if using a 95% confidence level?
113. {College Dean Narrative} What sample size should be used if we wish to estimate the population
mean to within 10 units?
114. A research firm has been contracted to estimate the mean weekly family expenditure on clothes. He
believes that the standard deviation of the weekly expenditure is $125. Determine with 99%
confidence the number of families that must be sampled to estimate the mean weekly family
expenditure on clothes to within $15.
115. How large a sample of federal employees in Nevada should be taken if we want to estimate with 98%
confidence the mean salary to within $2,000. The population standard deviation is assumed to be
$10,500.
116. An electronics retailer is interested in studying the incomes of consumers in a particular area. The
population standard deviation is known to be $1,000. What sample size would the researcher need to
use for a 95% confidence interval if the difference between UCL and LCL should not be more than
$100?
117. An engineer for an electric fencing company is interested in the mean length of wires being cut
automatically by machine. The desired length of the wires is 12 feet. It is known that the standard
deviation in the cutting length is 0.15 feet. Suppose the engineer decided to estimate the mean length
to within 0.025 with 99% confidence. What sample size would be needed?