a.
p-value =
b.
p-value <
c.
p-value >
d.
−
< p-value <
53. If the p value is less than
in a two-tail test:
a.
the null hypothesis should not be rejected.
b.
the null hypothesis should be rejected.
c.
a one-tail test should be used.
d.
No conclusion should be reached.
54. If a marketer wishes to determine whether there is evidence that average family income in a
community exceeds $32,000:
a.
either a one-tail or two-tail test could be used with equivalent results.
b.
a one-tail test should be utilized.
c.
a two-tail test should be utilized.
d.
None of these choices.
55. We have created a 95% confidence interval for
with the results (10, 25). What conclusion will we
make if we test H0:
= 26 vs. H1:
26 at
= 0.025?
a.
Reject H0 in favor of H1
b.
Accept H0 in favor of H1
c.
Fail to reject H0 in favor of H1
d.
We cannot tell from the information given.
56. The rejection region for testing H0:
= 100 vs. H1:
100, at the 0.05 level of significance is:
a.
| z | < 0.95
b.
| z | > 1.96
c.
z > 1.65
d.
z < 2.33
57. The owner of a local nightclub has recently surveyed a random sample of n = 300 customers of the
club. She would now like to determine whether or not the mean age of her customers is over 35. If so,
she plans to alter the entertainment to appeal to an older crowd. If not, no entertainment changes will
be made. Suppose she found that the sample mean was 35.5 years and the population standard
deviation was 5 years. What is the p-value associated with the test statistic?
a.
0.9582
b.
1.7300
c.
0.0418
d.
0.0836
58. It is possible to directly compare the results of a confidence interval estimate to the results obtained by
testing a null hypothesis if:
a.
a two-tail test for
is used.
b.
a one-tail test for
is used.
c.
a two-tail test for is used.
d.
a one-tail test for is used.
59. The rejection region for testing H0:
= 80 vs. H1:
80, at the 0.10 level of significance is:
a.
z > 1.96
b.
z < 0.90
c.
z > 1.28
d.
z < −1.28
60. We have created a 90% confidence interval for
with the result (25, 32). What conclusion will we
make if we test H0:
= 28 vs. H1:
28 at
= 0.10?
a.
Reject H0 in favor of H1.
b.
Accept H0 in favor of H1.
c.
Fail to reject H0 in favor of H1.
d.
We cannot tell from the information given.
61. The numerical quantity computed from the data that is used in deciding whether to reject H0 is the:
a.
significance level.
b.
critical value.
c.
test statistic.
d.
parameter.
62. The owner of a local Karaoke Bar has recently surveyed a random sample of n = 300 customers of the
bar. She would now like to determine whether or not the mean age of her customers is over 35. If so,
she plans to alter the entertainment to appeal to an older crowd. If not, no entertainment changes will
be made. If she wants to be 99% confident in her decision, what rejection region she use if the
population standard deviation
is known?
a.
Reject H0 if z < −2.33
b.
Reject H0 if z < −2.58
c.
Reject H0 if z > 2.33
d.
Reject H0 if z > 2.58
63. There are two approaches to making a decision in a hypothesis test once the test statistic has been
calculated. One approach is the ____________________ method. The other approach is the p-value
method.
ANS:
64. There are two approaches to making a decision in a hypothesis test once the test statistic has been
calculated. One approach is the rejection region method. The other approach is the
____________________ method.
65. The ____________________ is a range of values such that if the test statistic falls into that range we
reject the null hypothesis in favor of the alternative
hypothesis.
66. The probability of a test statistic falling in the rejection region is equal to the value of
____________________.
67. When a null hypothesis is rejected, the test is said to be statistically ____________________ at level
.
68. If the conclusion of a hypothesis test is that a statistically significant result was found, then the null
hypothesis ____________________ (was/was not) rejected.
69. The ____________________ of a test is the probability of observing a test statistic at least as extreme
as the one from your sample, given that H0 is true.
70. You reject H0 if the p–value of your hypothesis is ____________________ than the significance level.
71. The ____________________ is a measure of the amount of statistical evidence that supports the
alternative hypothesis.
72. If we do not reject the null hypothesis, we conclude that there ____________________(is/is not)
enough statistical evidence to infer that the alternative hypothesis is true.
Production Filling
A production filling operation has a historical standard deviation of 6 ounces. When in proper
adjustment, the mean filling weight for the production process is 50 ounces. A quality control
inspector periodically selects at random 36 containers and uses the sample mean filling weight to see if
the process is in proper adjustment.
73. {Production Filling Narrative} State the null and alternative hypotheses.
74. {Production Filling Narrative} Using a standardized test statistic, test the hypothesis at the 5% level of
significance if the sample mean filling weight is 48.6 ounces.
75. {Production Filling Narrative} Develop a 95% confidence interval and use it to test the hypothesis.
76. A social researcher claims that the average adult listens to the radio less than 26 hours per week. He
collects data on 25 individuals’ radio listening habits and finds that the mean number of hours that the
25 people spent listening to the radio was 22.4 hours. If the population standard deviation is known to
be eight hours, can we conclude at the 1% significance level that he is right?
77. A random sample of 100 observations from a normal population whose standard deviation is 50
produced a mean of 75. Does this statistic provide sufficient evidence at the 5% level of significance to
infer that the population mean is not 80?
78. In testing the hypotheses H0:
= 50 vs. H1:
< 50, we found that the standardized test statistic is z =
−1.59. Calculate the p-value, and state your conclusion if
= .025.
79. Suppose that 10 observations are drawn from a normal population whose variance is 64. The
observations are: 58, 62, 45, 50, 59, 65, 39, 40, 41, and 52. Test at the 10% level of significance to
determine if there is enough evidence to conclude that the population mean is greater than 45.
80. Suppose that 9 observations are drawn from a normal population whose standard deviation is 2. The
observations are: 15, 9, 13, 11, 8, 12, 11, 7, and 10. At 95% confidence, you want to determine
whether the mean of the population from which this sample was taken is significantly different from
10.
a.
State the null and alternative hypotheses.
b.
Compute the value of the test statistic.
c.
Compute the p-value.
d.
Interpret the results.
ANS:
a.
b.
c.
0.1587 2 = 0.3174.
d.
Cannot reject H0. Not enough evidence to say the mean is other than 10.
81. Determine the p-value associated with each of the following values of the standardized test statistic z,
and state your conclusion.
a.
two-tail test, with z = 1.50, and
= .10
b.
one-tail test, with z = 1.05, and
= .05
c.
one-tail test, with z = −2.40, and
= .01
0.1336, fail to reject H0
b.
0.1469, fail to reject H0
0.0082, reject H0
Watching the News
A researcher claims viewers spend an average of 40 minutes per day watching the news. You think the
average is higher than that. In testing your hypotheses H0:
= 40 vs. H1:
> 40, the following
information came from your random sample of viewers: = 42 minutes, n = 25. Assume
= 5.5, and
= 0.10.
82. {Watching the News Narrative} Calculate the value of the test statistic.
83. {Watching the News Narrative} Set up the rejection region.
84. {Watching the News Narrative} Determine the p-value.
85. {Watching the News Narrative} Interpret the result.
LSAT Scores
The Admissions officer for the graduate programs at the University of Pennsylvania believes that the
average score on the LSAT exam at his university is significantly higher than the national average of
1,300. An accepted standard deviation for LSAT scores is 125. A random sample of 25 scores had an
average of 1,375.
86. {LSAT Scores Narrative} State the appropriate null and alternative hypotheses.
87. {LSAT Scores Narrative} Calculate the value of the test statistic and set up the rejection region at the
0.025 level. What is your conclusion?
88. {LSAT Scores Narrative} Calculate the p-value.
89. {LSAT Scores Narrative} Use the p-value to test the hypotheses.
90. With the following p–values, would you reject or fail to reject the null hypothesis? Comment on the
statistical significance of each result. (Assume you normally reject H0 at level 0.08.)
a.
p-value = 0.0025
b.
p-value = 0.0328
c.
p-value = 0.0795
d.
p-value = 0.1940
ANS:
a.
b.
There is strong evidence to reject H0. The test is deemed to be significant.
c.
d.
There is no evidence to reject H0. The test is not statistically significant.
Marathon Runners
A researcher wants to study the average miles run per day for marathon runners. In testing the
hypotheses: H0:
= 25 miles vs. H1:
25 miles, a random sample of 36 marathon runners drawn
from a normal population whose standard deviation is 10, produced a mean of 22.8 miles weekly.
91. {Marathon Runners Narrative} Compute the value of the test statistic and specify the rejection region
associated with 5% significance level.
92. {Marathon Runners Narrative} Compute the p-value.
93. {Marathon Runners Narrative} What can we conclude at the 5% significance level regarding the null
hypothesis?
94. {Marathon Runners Narrative} Develop a 95% confidence interval estimate of the population mean.
95. {Marathon Runners Narrative} Explain briefly how to use the confidence interval to test the
hypothesis.
Toaster Oven
An appliance manufacturer claims to have developed a new toaster oven that consumes an average of
no more than 250 W. From previous studies, it is believed that power consumption for toaster ovens is
normally distributed with a standard deviation of 18 W. A consumer group suspects the actual average
is more than 250 W. They take a sample of 20 toaster ovens and calculate the average consumption to
be 260 W.
96. {Toaster Oven Narrative} What is the parameter of interest in this situation?
97. {Toaster Oven Narrative} State the appropriate hypotheses for the consumer group to do their test.
ANS:
98. {Toaster Oven Narrative} For a test with a level of significance of 0.05, determine the critical value.
99. {Toaster Oven Narrative} What is the value of the test statistic?
100. {Toaster Oven Narrative} Calculate the p-value of the test.
101. {Toaster Oven Narrative} What is the conclusion from the hypothesis test using
= .05?
102. There is a direct relationship between the power of a test and the probability of a Type II error.
103. The power of the test refers to the probability of rejecting a false null hypothesis.
104. For a given level of significance, if the sample size is increased, the power of the test will increase.
105. If a sample size is increased at a given
level, the probability of committing a Type II error is
increased.
106. The power of a test is the probability that a true null hypothesis will be rejected.
107. For a given level of significance, if the sample size is increased, the probability of committing a Type
II error will increase.
108. For a given sample size, the probability of committing a Type II error will increase when the
probability of committing a Type I error is reduced.
109. The operating characteristic curve plots the values of
(the probability of committing a Type II error)
versus the values of the population mean
.
110. One way of expressing how well a test performs is to report its power—the probability of detecting a
false null hypothesis.
111. As the alternative value of
increases, so does the power of the test.
112. The power of a test is measured by its capability of:
a.
rejecting a null hypothesis that is true.
b.
not rejecting a null hypothesis that is true.
c.
rejecting a null hypothesis that is false.
d.
not rejecting a null hypothesis that is false.
113. The power of a test is denoted by:
a.
b.
c.
1 −
d.
1 −
ANS:
114. For a given level of significance, if the sample size increases, the probability of a Type II error will:
a.
remain the same.
b.
increase.
c.
decrease.
d.
be equal to 1.0 regardless of
.
115. For a given sample size n, if the level of significance
is decreased, the power of the test will:
a.
increase.
b.
decrease.
c.
remain the same.
d.
Not enough information to tell.
116. For a given level of significance
, if the sample size n is increased, the probability of a Type II error
will:
a.
decrease.
b.
increase.
c.
remain the same.
d.
Not enough information to tell.
117. If the probability of committing a Type I error for a given test is decreased, then for a fixed sample
size n, the probability of committing a Type II error will:
a.
decrease.
b.
increase.
c.
stay the same.
d.
Not enough information to tell.
118. If we want to compute the probability of a Type II error, which of the following statements is false?
a.
We need to know the significance level
.
b.
We need to know the sample size n.
c.
We need to know the alternative value of the population mean
.
d.
All of these choices are true.
119. Which of the following statements is false regarding the operating characteristic (OC) curve?
a.
The OC curve plots the values of
versus the values of
.
b.
The OC curve plots the values of
versus the values of
.
c.
The OC curve can be useful in selecting a sample size n.
d.
None of these choices.
120. For a given level of significance, if the sample size is increased, the probability of committing a Type
II error will ____________________.
121. For a given level of significance, if the sample size is increased, the power of the test will
____________________.
122. To calculate the probability of a(n) ____________________ error you need to specify a value of
other than the one given in the null hypothesis.
123. Probabilities for Type I and Type II errors are actually ____________________ probabilities.
124. By ____________________ the significance level, you increase the probability of a Type II error.
125. The ____________________ of a test is a measure of its performance.
126. To increase the power of a test, ____________________ the sample size.
127. The power of a test plus the probability of a Type II error equals ____________________.
128. A(n) ____________________ characteristic curve plots the probability of a Type II error for various
alternative values of ____________________.
129. As the sample size increases, the operating characteristic curves drop down to zero at a(n)
____________________ rate.
130. To test the hypotheses: H0:
= 40 vs. H1:
40, we draw a random sample of size 16 from a normal
population whose standard deviation is 5. If we set
= 0.01, find the probability of committing a Type
II error when
= 37.
131. Calculate the probability of a Type II error for the hypothesis test: H0:
= 50 vs. H1:
> 50, given that
= 55,
= 0.05,
= 10, and n = 16.
Rechargeable Batteries
A researcher wants to study the average lifetime of a certain brand of rechargeable batteries (in hours).
In testing the hypotheses, H0:
= 950 hours vs. H1:
950 hours, a random sample of 25
rechargeable batteries is drawn from a normal population whose standard deviation is 200 hours.
132. {Rechargeable Batteries Narrative} Calculate
, the probability of a Type II error when
= 1000 and
= 0.10.
133. {Rechargeable Batteries Narrative} Calculate the power of the test when
= 1000 and
= 0.10.
134. {Rechargeable Batteries Narrative} Interpret the meaning of the power of the test.
135. {Rechargeable Batteries Narrative} Recalculate
if n is increased from 25 to 40.
136. {Rechargeable Batteries Narrative} Review the results of the previous questions. What is the effect of
increasing the sample size on the value of
?
137. {Rechargeable Batteries Narrative} Recalculate
if
is lowered from 0.10 to 0.05.
138. {Rechargeable Batteries Narrative} Review the results of the previous questions. What is the effect of
decreasing the significance level on the value on
?
139. During the last energy crisis, a government official claimed that the average car owner refills the tank
when there is more than 3 gallons left. To check the claim, 10 cars were surveyed as they entered a gas
station. The amount of gas remaining before refill was measured and recorded as follows (in gallons):
3, 5, 3, 2, 3, 3, 2, 6, 4, and 1. Assume that the amount of gas remaining in tanks is normally distributed
with a standard deviation of 1 gallon. Compute the probability of a Type II error and the power of the
test if the true average amount of gas remaining in tanks is 3.5 gallons and
= 0.10.