Download Statistics: Informed Decisions Using Data, 4e (Sullivan)

Ch. 10 Hypothesis Tests Regarding a Parameter
10.1 The Language of Hypothesis Testing
1 Determine the null and alternative hypotheses.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
The null and alternative hypotheses are given. Determine whether the hypothesis test is left -tailed, right-tailed, or
two-tailed and the parameter that is being tested.
1) H : μ = 8.6 0
H : μ ≠ 8.6
1
C) Right-tailed, μ
D) Left-tailed, x
A) Two-tailed, μ
B) Two-tailed, x
2) H
H
3) H
H
0
: p = 0.82 : p > 0.82 1
A) Right-tailed, p
0
^
^
B) Left-tailed, p
C) Right-tailed, p
D) Left-tailed, p
B) Right-tailed, μ
C) Right-tailed, σ
D) Left-tailed, s
: σ = 8.6 : σ < 8.6
1
A) Left-tailed, σ
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
4) The mean annual return for an employeeʹs IRA is at most 3.4 percent. Write the null and alternative
hypotheses.
5) The mean age of lawyers in New York is 50.7 years. Write the null and alternative hypotheses.
6) The mean repair bill of cars is greater than $110. Write the null and alternative hypotheses.
7) The mean utility bill in one city during the summer was less than $90. Write the null and alternative
hypotheses.
8) The mean annual return for an employeeʹs IRA is at most 4.7 percent. Write the null and alternative
hypotheses.
9) A popular referendum on the ballot is favored by more than half of the voters. Write the null and alternative
hypotheses.
10) The owner of an outdoor store recommends against buying the new model of one brand of GPS receivers
because they vary more than the old model, which had a standard deviation of 50 meters. Write the null and
alternative hypotheses.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
11) A ______________ is a statement or claim regarding a characteristic of one or more populations.
A) hypothesis
B) conclusion
C) conjecture
D) fact
12) The ______________ hypothesis contains the ʺ =ʺ sign.
A) null
B) alternative
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C) explanatory
D) conditional
13) A hypothesis test is a ʺtwo-tailedʺ if the alternative hypothesis contains a _______ sign.
A) ≠
B) +
C) <
D) >
2 Explain Type I and Type II errors.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
1) The mean age of judges in Los Angeles is 48.6 years. Identify the type I and type II errors for the hypothesis test
of this claim.
2) The mean cost of textbooks for one class is greater than $140. Identify the type I and type II errors for the
hypothesis test of this claim.
3) The mean monthly cell phone bill for one household was less than $ 93. Identify the type I and type II errors for
the hypothesis test of this claim.
4) A referendum for an upcoming election is favored by more than half of the voters. Identify the type I and type
II errors for the hypothesis test of this claim.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
5) A local tennis pro-shop strings tennis rackets at the tension (pounds per square inch) requested by the
customer. Recently a customer made a claim that the pro-shop consistently strings rackets at lower tensions, on
average, than requested. To support this claim, the customer asked the pro shop to string 12 new rackets at 53
psi. Upon receiving the rackets, the customer measured the tension of each and calculated the following
summary statistics: x = 51 psi, s = 2.4 psi. In order to conduct the test, the customer selected a significance level
of α = .05. Interpret this value.
A) The probability of concluding that the true mean is less than 53 psi when in fact it is equal to 53 psi is only
.05.
B) The smallest value of α that you can use and still reject H0 is 0.05.
C) The probability of making a Type II error is 0.95.
D) There is a 5% chance that the sample will be biased.
6) True or False: If I specify β to be equal to 0.36, then the value of α must be 0.64.
A) False
B) True
7) What is the probability associated with not making a Type II error?
A) (1 - β)
B) α
C) β
D) (1 - α)
8) We never conclude ʺAccept H0 ʺ in a test of hypothesis. This is because:
A) β = p(Type II error) is not known.
C) The rejection region is not known.
B) α is the probability of a Type I error.
D) The p-value is not small enough.
9) If we reject the null hypothesis when the null hypothesis is true, then we have made a
A) Type I error
B) Type II error
C) Correct decision
D) Type α error
10) If we do not reject the null hypothesis when the null hypothesis is in error, then we have made a
A) Type II error
B) Type I error
C) Correct decision
D) Type β error
11) The level of significance, α, is the probability of making a
A) Type I error
B) Type II error
C) Correct decision
12) True or False: Type I and Type II errors are independent events.
A) False
B) True
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D) Type β error
3 State conclusions to hypothesis tests.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1) The mean age of principals in a local school district is 50.2 years. If a hypothesis test is performed, how should
you interpret a decision that rejects the null hypothesis?
A) There is sufficient evidence to reject the claim μ = 50.2.
B) There is not sufficient evidence to reject the claim μ = 50.2.
C) There is sufficient evidence to support the claim μ = 50.2.
D) There is not sufficient evidence to support the claim μ = 50.2.
2) The mean age of professors at a university is 57.9 years. If a hypothesis test is performed, how should you
interpret a decision that fails to reject the null hypothesis?
A) There is not sufficient evidence to reject the claim μ = 57.9.
B) There is sufficient evidence to reject the claim μ = 57.9.
C) There is sufficient evidence to support the claim μ = 57.9.
D) There is not sufficient evidence to support the claim μ = 57.9.
3) The mean age of professors at a university is greater than 58.3 years. If a hypothesis test is performed, how
should you interpret a decision that rejects the null hypothesis?
A) There is sufficient evidence to support the claim μ > 58.3.
B) There is sufficient evidence to reject the claim μ > 58.3.
C) There is not sufficient evidence to reject the claim μ > 58.3.
D) There is not sufficient evidence to support the claim μ > 58.3.
4) The mean age of judges in Dallas is greater than 50.2 years. If a hypothesis test is performed, how should you
interpret a decision that fails to reject the null hypothesis?
A) There is not sufficient evidence to support the claim μ > 50.2.
B) There is sufficient evidence to reject the claim μ > 50.2.
C) There is not sufficient evidence to reject the claim μ > 50.2.
D) There is sufficient evidence to support the claim μ > 50.2.
5) The mean monthly gasoline bill for one household is greater than $150. If a hypothesis test is performed, how
should you interpret a decision that rejects the null hypothesis?
A) There is sufficient evidence to support the claim μ > $150.
B) There is sufficient evidence to reject the claim μ > $150.
C) There is not sufficient evidence to reject the claim μ > $150.
D) There is not sufficient evidence to support the claim μ > $150.
6) The mean monthly gasoline bill for one household is greater than $160. If a hypothesis test is performed, how
should you interpret a decision that fails to reject the null hypothesis?
A) There is not sufficient evidence to support the claim μ > $160.
B) There is sufficient evidence to reject the claim μ > $160.
C) There is not sufficient evidence to reject the claim μ > $160.
D) There is sufficient evidence to support the claim μ > $160.
7) The mean number of rushing yards for one NFL team was less than 99 yards per game. If a hypothesis test is
performed, how should you interpret a decision that rejects the null hypothesis?
A) There is sufficient evidence to support the claim μ < 99.
B) There is sufficient evidence to reject the claim μ < 99.
C) There is not sufficient evidence to reject the claim μ < 99.
D) There is not sufficient evidence to support the claim μ < 99.
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8) The mean number of rushing yards for one NFL team was less than 109 yards per game. If a hypothesis test is
performed, how should you interpret a decision that fails to reject the null hypothesis?
A) There is not sufficient evidence to support the claim μ < 109.
B) There is sufficient evidence to reject the claim μ < 109.
C) There is not sufficient evidence to reject the claim μ < 109.
D) There is sufficient evidence to support the claim μ < 109.
9) The dean of a major university claims that the mean number of hours students study at her University (per
day) is at most 3.6 hours. If a hypothesis test is performed, how should you interpret a decision that rejects the
null hypothesis?
A) There is sufficient evidence to reject the claim μ ≤ 3.6.
B) There is not sufficient evidence to reject the claim μ ≤ 3.6.
C) There is sufficient evidence to support the claim μ ≤ 3.6.
D) There is not sufficient evidence to support the claim μ ≤ 3.6.
10) The dean of a major university claims that the mean number of hours students study at her University (per
day) is at most 4.4 hours. If a hypothesis test is performed, how should you interpret a decision that fails to
reject the null hypothesis?
A) There is not sufficient evidence to reject the claim μ ≤ 4.4.
B) There is sufficient evidence to reject the claim μ ≤ 4.4.
C) There is sufficient evidence to support the claim μ ≤ 4.4.
D) There is not sufficient evidence to support the claim μ ≤ 4.4.
11) A candidate for state representative of a certain state claims to be favored by at least half of the voters. If a
hypothesis test is performed, how should you interpret a decision that rejects the null hypothesis?
A) There is sufficient evidence to reject the claim p ≥ 0.5.
B) There is not sufficient evidence to reject the claim p ≥ 0.5.
C) There is sufficient evidence to support the claim p ≥ 0.5.
D) There is not sufficient evidence to support the claim p ≥ 0.5.
12) A candidate for state representative of a certain state claims to be favored by at least half of the voters. If a
hypothesis test is performed, how should you interpret a decision that fails to reject the null hypothesis?
A) There is not sufficient evidence to reject the claim p ≥ 0.5.
B) There is sufficient evidence to reject the claim p ≥ 0.5.
C) There is sufficient evidence to support the claim p ≥ 0.5.
D) There is not sufficient evidence to support the claim p ≥ 0.5.
10.2 Hypothesis Tests for a Population Proportion
1 Explain the logic of hypothesis testing.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1) Find the critical value for a left-tailed test with α = 0.025.
A) -1.96
B) -2.24
2) Find the critical value for a two-tailed test with α = 0.08.
B) ±1.405
A) ±1.75
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C) -1.645
D) -2.575
C) ±2.65
D) ±1.96
3) Suppose you want to test the claim that μ = 3.5. Given a sample size of n = 48 and a level of significance of
α = 0.01, when should you reject H0 ?
A) Reject H0 if the standardized test statistic is greater than 2.575 or less than -2.575.
B) Reject H0 if the standardized test statistic is greater than 2.33 or less than -2.33.
C) Reject H0 if the standardized test statistic is greater than 1.96 or less than -1.96
D) Reject H0 if the standardized test statistic is greater than 1.645 or less than -1.645
4) Suppose you want to test the claim that μ > 25.6. Given a sample size of n = 53 and a level of significance of
α = 0.01, when should you reject H0 ?
A) Reject H0 if the standardized test statistic is greater than 2.33.
B) Reject H0 if the standardized test statistic is greater than 1.96.
C) Reject H0 if the standardized test statistic is greater than 2.575.
D) Reject H0 if the standardized test statistic is greater than 1.645.
5) Suppose you want to test the claim that μ ≥ 65.4. Given a sample size of n = 35 and a level of significance of
α = 0.01, when should you reject H0 ?
A) Reject H0 if the standardized test statistic is less than -2.33.
B) Reject H0 if the standardized test is less than -2.575.
C) Reject H0 if the standardized test statistic is less than -1.96.
D) Reject H0 if the standardized test statistic is less than -1.28.
6) When the results of a hypothesis test are determined to be statistically significant, then we _______________ the
null hypothesis.
A) reject
B) fail to reject
C) polarize
D) compartmentalize
2 Test hypotheses about a population proportion.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1) The business college computing center wants to determine the proportion of business students who have
personal computers (PCʹs) at home. If the proportion exceeds 35%, then the lab will scale back a proposed
enlargement of its facilities. Suppose 300 business students were randomly sampled and 85 have PCʹs at home.
Find the rejection region for this test using α = 0.10.
A) Reject H0 if z > 1.28.
B) Reject H0 if z < -1.28.
C) Reject H0 if z > 1.645 or z < -1.645.
D) Reject H0 if z = 1.28.
2) The business college computing center wants to determine the proportion of business students who have
personal computers (PCʹs) at home. If the proportion differs from 35%, then the lab will modify a proposed
enlargement of its facilities. Suppose a hypothesis test is conducted and the test statistic is 2.6. Find the P-value
for a two-tailed test of hypothesis.
A) 0.0094
B) 0.0047
C) 0.4906
D) 0.4953
3) The business college computing center wants to determine the proportion of business students who have
personal computers (PCʹs) at home. If the proportion exceeds 30%, then the lab will scale back a proposed
enlargement of its facilities. Suppose 300 business students were randomly sampled and 65 have PCʹs at home.
What assumptions are necessary for this test to be satisfied?
A) No assumptions are necessary.
B) The sample variance equals the population variance.
C) The population has an approximately normal distribution.
D) The sample mean equals the population mean.
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4) A survey claims that 9 out of 10 doctors (i.e., 90%) recommend brand Z for their patients who have children. To
test this claim against the alternative that the actual proportion of doctors who recommend brand Z is less than
90%, a random sample of 100 doctors results in 83 who indicate that they recommend brand Z. The test statistic
in this problem is approximately (round to the nearest hundredth):
A) -2.33
B) 2.33
C) -1.83
D) -1.99
5) A survey claims that 9 out of 10 doctors (i.e., 90%) recommend brand Z for their patients who have children. To
test this claim against the alternative that the actual proportion of doctors who recommend brand Z is less than
90%, a random sample of doctors was taken. Suppose the test statistic is z = -2.23. Can we conclude that H0
should be rejected at the a) α =0.10, b) α = 0.05, and c) α = 0.01 level?
A) a) yes; b) yes; c) no
B) a) yes; b) yes; c) yes
C) a) no; b) no; c) no
D) a) no; b) no; c) yes
6) A nationwide survey claimed that at least 55% of parents with young children condone spanking their child as
a regular form of punishment. In a random sample of 100 parents with young children, how many would need
to say that they condone spanking as a form of punishment in order to refute the claim at α = 0.5?
A) You would need 46 or less parents to support spanking to refute the claim.
B) You would need exactly 46 parents to support spanking to refute the claim.
C) You would need 47 or less parents to support spanking to refute the claim.
D) You would need more than 46 parents to support spanking to refute the claim.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
7) A method currently used by doctors to screen women for possible breast cancer fails to detect cancer in 15% of
the women who actually have the disease. A new method has been developed that researchers hope will be
able to detect cancer more accurately. A random sample of 70 women known to have breast cancer were
screened using the new method. Of these, the new method failed to detect cancer in 8. Calculate the test
statistic used by the researchers for this test of hypothesis. Round to the nearest thousandth.
8) A method currently used by doctors to screen women for possible breast cancer fails to detect cancer in 15% of
the women who actually have the disease. A new method has been developed that researchers hope will be
able to detect cancer more accurately. A random sample of 80 women known to have breast cancer were
screened using the new method. Of these, the new method failed to detect cancer in 9. Is the sample size
sufficiently large in order to conduct this test of hypothesis? Explain. Round to the nearest thousandth.
9) Increasing numbers of businesses are offering child-care benefits for their workers. However, one union claims
that more than 85% of firms in the manufacturing sector still do not offer any child-care benefits to their
workers. A random sample of 430 manufacturing firms is selected, and only 32 of them offer child-care
benefits. Specify the rejection region that the union will use when testing at α = 0.10.
10) Increasing numbers of businesses are offering child-care benefits for their workers. However, one union claims
that more than 80% of firms in the manufacturing sector still do not offer any child-care benefits to their
workers. A random sample of 390 manufacturing firms is selected and asked if they offer child -care benefits.
Suppose the P-value for this test was reported to be p = 0.1112. State the conclusion of interest to the union.
Use α = 0.10.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
11) In a sample, 76 people or 38% of the people in the sample said that the mayor should be prosecuted for
misconduct. How many people where in the sample?
A) 200
B) 29
C) 105
D) 50
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12) Determine the critical value, z 0 , to test the claim about the population proportion p = 0.250 given n = 48 and
^
p = 0.231. Use α = 0.01.
A) ±2.575
B) ±1.96
C) ±1.645
D) ±2.33
13) Determine the standardized test statistic, z, to test the claim about the population proportion p ≥ 0.132 given
^
n = 48 and p = 0.11. Use α = 0.05.
A) -0.45
B) -1.645
C) -0.568
D) -0.124
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
^
14) Test the claim about the population proportion p ≥ 0.700 given n = 50 and p = 0.612. Use α = 0.10.
15) Fifty percent of registered voters in a congressional district are registered Democrats. The Republican candidate
takes a poll to assess his chances in a two-candidate race. He polls 1200 potential voters and finds that 621 plan
to vote for the Democratic candidate. Does the Republican candidate have a chance to win? Use α = 0.05.
16) An airline claims that the no-show rate for passengers is less than 5%. In a sample of 420 randomly selected
^
reservations, 19 were no-shows. At α = 0.01, test the airlineʹs claim. Round p to the nearest thousandth when
calculating the test statistic.
17) A recent study claimed that at least 15% of junior high students are overweight. In a sample of 160 students, 18
were found to be overweight. At α = 0.05, test the claim.
18) The engineering school at a major university claims that 20% of its graduates are women. In a graduating class
^
of 210 students, 58 were females. Does this suggest that the school is believable? Use α = 0.05. Round p to the
nearest ten-thousandth when calculating the test statistic.
19) A coin is tossed 1000 times and 570 heads appear. At α = 0.05, test the claim that this is not a biased coin.
20) In one city, 25 out of 100 randomly sampled teenagers say that they smoke.
(a) Consider the hypotheses H0 : p = 0.22 versus H1 : p > 0.22. Explain what the researcher would be testing.
Perform the test at the α = 0.05 level of significance. Write a conclusion for the test. Round the test statistic to
the nearest hundredth.
(b) Repeat part (a) for the hypotheses H0 : p = 0.24 versus H1 : p > 0.24.
(c) Based on your results in parts (a) and (b), write a few sentences that explain the difference between
ʺacceptingʺ the statement in the null hypothesis versus ʺnot rejectingʺ the statement in the null hypothesis.
3 Test hypotheses about a population proportion using the binomial probability distribution.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
1) According to a national statistics bureau, 3.7% of males living in the Southwest were retired exterminators. A
researcher believes that the percentage has increased since then. She randomly selects 250 males in the
Southwest and finds that 4 of them are retired exterminators. Test this researcherʹs claim at the α = 0.1 level of
significance.
2) According to a prestigious historical society, in 1999, 7.2% of recent high school graduates believe that the
Romans invented mayonnaise. A classics scholar believes that the percentage has increased since then. He
randomly selects 125 recent high school graduates and finds that 17 of them believe in the Roman invention of
mayonnaise. Test this researcherʹs claim at the α = 0.01 level of significance.
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3) According to a local chamber of commerce, in 1993, 5.9% of local area residents owned more than five cars. A
local car dealer claims that the percentage has increased. He randomly selects 180 local area residents and finds
that 12 of them own more than five cars. Test this car dealerʹs claim at the α = 0.05 level of significance.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
4) An event is considered unusual if the probability of observing the event is
A) less than 0.05
B) less than 0.025
C) less than 0.10
D) greater than 0.95
10.3 Hypothesis Tests for a Population Mean
1 Test hypotheses about a mean.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the critical value.
1) Determine the critical value for a right-tailed test of a population mean at the α = 0.001 level of significance
with 15 degrees of freedom.
A) 3.733
B) 2.602
C) -3.733
D) 3.787
2) Determine the critical value for a left-tailed test of a population mean at the α = 0.005 level of significance
based on a sample size of n = 30.
A) -2.756
B) -1.699
C) 2.756
D) 2.750
3) Determine the critical values for a two-tailed test of a population mean at the α = 0.01 level of significance
based on a sample size of n = 29.
A) ±2.763
B) ±2.756
C) ±2.467
D) ±2.462
Provide an appropriate response.
4) A relative frequency histogram for the sale prices of homes sold in one city during 2010 is shown below. Based
on the histogram, is a large sample necessary to conduct a hypothesis test about the mean sale price? If so,
why?
A) Yes; data do not appear to be normally distributed but skewed right.
B) Yes; data do not appear to be normally distributed but skewed left.
C) No; data appear to be normally distributed.
D) Yes; data do not appear to be normally distributed but bimodal.
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5) The ages of a group of patients being treated at one hospital for osteoporosis are summarized in the frequency
histogram below. Based on the histogram, is a large sample necessary to conduct a hypothesis test about the
mean age? If so, why?
A) Yes; data do not appear to be normally distributed but skewed left.
B) Yes; data do not appear to be normally distributed but skewed right.
C) Yes; data do not appear to be normally distributed but bimodal.
D) No; data appear to be normally distributed with no outliers.
6) The weekly salaries (in dollars) of randomly selected employees of a company are summarized in the boxplot
below. Based on the boxplot, is a large sample necessary to conduct a hypothesis test about the mean salary? If
so, why?
A) Yes; data do not appear to be normally distributed but skewed right.
B) Yes; data do not appear to be normally distributed but skewed left.
C) No; data appear to be normally distributed.
D) Yes; data contain outliers.
7) The weights (in ounces) of a sample of tomatoes of a particular variety are summarized in the boxplot below.
Based on the boxplot, is a large sample necessary to conduct a hypothesis test about the mean weight? If so,
why? A) Yes; data contain outliers.
B) Yes; data do not appear to be normally distributed but skewed left.
C) Yes; data do not appear to be normally distributed but skewed right.
D) No; data appear to be normally distributed with no outliers.
8) Find the standardized test statistic t for a sample with n = 12, x = 30.2, s = 2.2, and α = 0.01 if H0 : μ = 29. Round
your answer to three decimal places.
A) 1.890
B) 1.991
C) 2.132
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D) 2.001
9) Find the standardized test statistic t for a sample with n = 10, x = 7.9, s = 1.3, and α = 0.05 if H0 : μ ≥ 8.8. Round
your answer to three decimal places.
A) -2.189
B) -3.186
C) -3.010
D) -2.617
10) Find the standardized test statistic t for a sample with n = 15, x = 10.1, s = 0.8, and α = 0.05 if H0 : μ ≤ 9.8. Round
your answer to three decimal places.
A) 1.452
B) 1.728
C) 1.631
D) 1.312
11) Find the standardized test statistic t for a sample with n = 20, x = 13.3, s = 2.0, and α = 0.05 if H1 : μ < 13.7.
Round your answer to three decimal places.
A) -0.894
B) -0.872
C) -1.265
D) -1.233
12) Find the standardized test statistic t for a sample with n = 25, x = 14, s = 3, and α = 0.005 if H1 : μ > 13. Round
your answer to three decimal places.
A) 1.667
B) 1.997
C) 1.452
D) 1.239
13) Find the standardized test statistic t for a sample with n = 12, x = 15.7, s = 2.1, and α = 0.01 if H1 : μ ≠ 16.2.
Round your answer to three decimal places.
A) -0.825
B) -0.008
C) -0.037
D) -0.381
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
14) Use a t-test to test the claim μ = 10 at α = 0.01, given the sample statistics n = 12, x = 11.2, and s = 2.2. Round
the test statistic to the nearest thousandth.
15) Use a t-test to test the claim μ ≥ 9.7 at α = 0.05, given the sample statistics n = 10, x = 8.8, and s = 1.3. Round the
test statistic to the nearest thousandth.
16) Use a t-test to test the claim μ ≤ 8.9 at α = 0.05, given the sample statistics n = 15, x = 9.2, and s = 0.8. Round the
test statistic to the nearest thousandth.
17) Use a t-test to test the claim μ < 12.6 at α = 0.10, given the sample statistics n = 20, x = 12.2, and s = 2.0. Round
the test statistic to the nearest thousandth.
18) Use a t-test to test the claim μ > 34 at α = 0.005, given the sample statistics n = 25, x = 35, and s = 3. Round the
test statistic to the nearest thousandth.
19) Use a t-test to test the claim μ = 19.4 at α = 0.01, given the sample statistics n = 12, x = 18.9, and s = 2.1. Round
the test statistic to the nearest thousandth.
20) A local retailer claims that the mean waiting time is less than 9 minutes. A random sample of 20 waiting times
has a mean of 7.4 minutes with a standard deviation of 2.1 minutes. At α = 0.01, test the retailerʹs claim.
Assume the distribution is normally distributed. Round the test statistic to the nearest thousandth.
21) A manufacturer claims that the mean lifetime of its lithium batteries is 1400 hours. A homeowner selects 25 of
these batteries and finds the mean lifetime to be 1380 hours with a standard deviation of 80 hours. Test the
manufacturerʹs claim. Use α = 0.05. Round the test statistic to the nearest thousandth.
22) A shipping firm suspects that the mean life of a certain brand of tire used by its trucks is less than 40,000 miles.
To check the claim, the firm randomly selects and tests 18 of these tires and gets a mean lifetime of 39,300 miles
with a standard deviation of 1200 miles. At α = 0.05, test the shipping firmʹs claim. Round the test statistic to
the nearest thousandth.
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23) A local juice manufacturer distributes juice in bottles labeled 12 ounces. A government agency thinks that the
company is cheating its customers. The agency selects 20 of these bottles, measures their contents, and obtains a
sample mean of 11.7 ounces with a standard deviation of 0.7 ounce. Use a 0.01 significance level to test the
agencyʹs claim that the company is cheating its customers. Round the test statistic to the nearest thousandth.
24) A local group claims that the police issue at least 60 parking tickets a day in their area. To prove their point,
they randomly select two weeks. Their research yields the number of tickets issued for each day. The data are
listed below. At α = 0.01, test the groupʹs claim. Round the test statistic to the nearest thousandth.
70 48 41 68 69 55 70 57 60 83
32 60 72 58
25) A bank claims that the mean waiting time in line is less than 4.3 minutes. A random sample of 20 customers has
a mean of 4.1 minutes with a standard deviation of 0.8 minute. If α = 0.05, test the bankʹs claim using P-values.
26) A local hardware store claims that the mean waiting time in line is less than 3.5 minutes. A random sample of
20 customers has a mean of 3.7 minutes with a standard deviation of 0.8 minute. If α = 0.05, test the storeʹs
claim using P-values.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
27) A local eat-in pizza restaurant wants to investigate the possibility of starting to deliver pizzas. The owner of
the store has determined that home delivery will be successful if the average time spent on the deliveries does
not exceed 32 minutes. The owner has randomly selected 23 customers and has delivered pizzas to their homes
in order to test if the mean delivery time actually exceeds 32 minutes. Suppose the P-value for the test was
found to be 0.0297. State the correct conclusion.
A) At α = 0.025, we fail to reject H0 .
B) At α = 0.05, we fail to reject H0 .
D) At α = 0.03, we fail to reject H0 .
C) At α = 0.02, we reject H0 .
28) A local tennis pro-shop strings tennis rackets at the tension (pounds per square inch) requested by the
customer. Recently a customer made a claim that the pro-shop consistently strings rackets at lower tensions, on
average, than requested. To support this claim, the customer asked the pro shop to string 13 new rackets at 65
psi. Suppose the two-tailed P-value for the test described above (obtained from a computer printout) is 0.09.
Give the proper conclusion for the test. Use α = 0.05.
A) There is sufficient evidence to conclude that μ, the true mean tension of the rackets, is less than 65 psi.
B) Accept H0 and conclude that μ, the true mean tension of the rackets, equals 65 psi.
C) There is insufficient evidence to conclude that μ, the true mean tension of the rackets, is less than 65 psi.
D) Reject H0 and conclude that μ, the true mean tension of the rackets, equals 65 psi.
29) If we have a sample of 12 drawn from a normal population, then we would use as our test statistic
A) t0 with 11 degrees of freedom
B) z0 with 11 degrees of freedom
C) t0 with 12 degrees of freedom
D) z0 with 12 degrees of freedom
2 Understand the difference between statistical significance and practical significance.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1) True or False: Results that are statistically significant are always practically significant.
A) False
B) True
2) True or False: Results that are practically significant will always be statistically significant.
A) False
B) True
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10.4 Hypothesis Tests for a Population Standard Deviation
1 Test hypotheses about a population standard deviation.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the critical value(s).
1) Determine the critical value for a right-tailed test of a population standard deviation with 22 degrees of
freedom at the α = 0.05 level of significance.
A) 33.924
B) 32.671
C) 42.796
D) 12.338
2) Determine the critical value for a left-tailed test of a population standard deviation for a sample of size n = 21
at the α = 0.05 level of significance.
A) 10.851
B) 11.591
C) 31.41
D) 32.671
3) Determine the critical values for a two-tailed test of a population standard deviation for a sample of size n = 7
at the α = 0.1 level of significance.
A) 1.635, 12.592
B) 2.167, 14.067
C) 2.204, 10.645
D) 2.833, 12.017
Provide an appropriate response. Round the test statistic to the nearest thousandth.
4) Compute the standardized test statistic, χ 2 , to test the claim σ2 = 30.1 if n = 12, s 2 = 25.2, and α = 0.05.
A) 9.209
B) 12.961
C) 18.490
D) 0.492
5) Compute the standardized test statistic, χ 2 , to test the claim σ2 ≥ 7.2 if n = 15, s 2 = 6, and α = 0.05.
A) 11.667
B) 8.713
C) 12.823
D) 23.891
6) Compute the standardized test statistic, χ 2 , to test the claim σ2 ≤ 25.6 if n = 20, s 2 = 49.6, and α = 0.01.
A) 36.813
B) 9.322
C) 12.82
D) 33.41
7) Compute the standardized test statistic, χ 2 , to test the claim σ2 > 15.2 if n = 18, s 2 = 21.6, and α = 0.01.
A) 24.158
B) 28.175
C) 33.233
D) 43.156
8) Compute the standardized test statistic, χ 2 , to test the claim σ2 < 50.4 if n = 28, s 2 = 31.5, and α = 0.10.
A) 16.875
B) 14.324
C) 18.132
D) 21.478
9) Compute the standardized test statistic, χ 2 to test the claim σ2 ≠ 20.4 if n = 10, s 2 = 22.5, and α = 0.01.
A) 9.926
B) 3.276
C) 4.919
D) 12.008
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
10) Test the claim that σ2 = 30.1 if n = 12, s 2 = 25.2 and α = 0.05. Assume that the population is normally
distributed.
11) Test the claim that σ2 ≥ 1.8 if n = 15, s 2 = 1.5, and α = 0.05. Assume that the population is normally distributed.
12) Test the claim that σ2 ≤ 28.8 if n = 20, s 2 = 55.8, and α = 0.01. Assume that the population is normally
distributed.
13) Test the claim that σ2 > 17.1 if n = 18, s 2 = 24.3, and α = 0.01. Assume that the population is normally
distributed.
14) Test the claim that σ2 < 50.4 if n = 28, s 2 = 31.5, and α = 0.10. Assume that the population is normally
distributed.
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15) Test the claim that σ2 ≠ 54.4 if n = 10, s 2 = 60, and α = 0.01. Assume that the population is normally distributed.
16) Test the claim that σ = 18.63 if n = 12, s = 17.1, and α = 0.05. Assume that the population is normally
distributed.
17) Test the claim that σ ≥ 9.38 if n = 15, s = 8.54, and α = 0.05. Assume that the population is normally distributed.
18) Test the claim that σ ≤ 5.37 if n = 20, s = 7.47, and α = 0.01. Assume that the population is normally distributed.
19) Test the claim that σ > 12.42 if n = 18, s = 14.76, and α = 0.01. Assume that the population is normally
distributed.
20) Test the claim that σ < 9.48 if n = 28, s = 7.48 and α = 0.10. Assume that the population is normally distributed.
21) Test the claim that σ ≠ 20.88 if n = 10, s = 21.92, and α = 0.01. Assume that the population is normally
distributed.
22) Listed below are the April utility bills (in dollars) for one neighborhood. Assuming that the data is normally
distributed, test the claim that the standard deviation for the data is $15. Use α = 0.01.
70 48 41 68 69 55 70 57 60 83
32 60 72 58
23) The commute times (in minutes) of 20 randomly selected adult males are listed below. Test the claim that the
variance is less than 6.25. Use α = 0.05. Assume the population is normally distributed.
70 72 71 70 69 73 69 68 70 71
67 71 70 74 69 68 71 71 71 72
24) A shipping firm suspects that the variance for a certain brand of tire used by its trucks is greater than 1,000,000.
To check the claim, the firm puts 101 of these tires on its trucks and gets a standard deviation of 1200 miles. At
α = 0.05, test the shipping firmʹs claim.
25) A brokerage firm needs information concerning the standard deviation of the account balances of its customers.
From previous information it was assumed to be $250. A random sample of 61 accounts was checked. The
standard deviation was $286.20. At α = 0.01, test the firmʹs assumption. Assume that the account balances are
normally distributed.
26) In one area, monthly incomes of technology related workers have a standard deviation of $650. It is believed
that the standard deviation of monthly incomes of non-technology workers is higher. A sample of 71
non-technology workers are randomly selected and found to have a standard deviation of $950. Test the claim
that non-technology workers have a higher standard deviation. Use α = 0.05.
27) A statistics professor at an all-menʹs college determined that the standard deviation of menʹs heights is 2.5
inches. The professor then randomly selected 41 female students from a nearby all -female college and found
the standard deviation to be 2.9 inches. Test the professorʹs claim that the standard deviation of female heights
is greater than 2.5 inches. Use α = 0.01.
28) A new gun-like apparatus has been devised to replace the needle in administering vaccines. The apparatus,
which is connected to a large supply of vaccine, can be set to inject different amounts of the serum, but the
variance in the amount of serum injected to a given person must not be greater than 0.0 8 to ensure proper
inoculation. A random sample of 25 injections was measured. Suppose the P-value for the test is p = 0.0085.
State the proper conclusion using α = 0.01.
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10.5 Putting It Together: Which Method Do I Use?
1 Determine the appropriate hypothesis test to perform.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
1) In 2010, 36% of adults in a certain country were morbidly obese. A health practitioner suspects that the percent
has changed since then. She obtains a random sample of 1042 adults and finds that 393 are morbidly obese. Is
^
this sufficient evidence to support the practitionerʹs claim at the α = 0.1 level of significance? Round p to five
decimal places when calculating the test statistic.
2) In 2010, the mean expenditure for auto insurance in a certain state was $806. An insurance salesperson in this
state believes that the mean expenditure for auto insurance is less today. She obtains a simple random sample
of 32 auto insurance policies and determines the mean expenditure to be $781 with a standard deviation of
$39.13. Is there enough evidence to support the claim that the mean expenditure for auto insurance is less than
the 2010 amount at the α = 0.05 level of significance?
3) A pharmaceutical company manufactures a 81-mg pain reliever. Company specifications include that the
standard deviation of the amount of the active ingredient must not exceed 2 mg. The quality -control manager
selects a random sample of 30 tablets from a certain batch and finds that the standard deviation is 2.4 mg.
Assume that the amount of the active ingredient is normally distributed. Test the claim that the standard
deviation of the amount of the active ingredient is greater than 2 mg at the α = 0.05 level of significance.
4) In 2010, the mean for the number of pets owned per household was 1.9. A poll of 1023 households conducted
this year reported the mean for the number of pets owned per household to be 1.8. Assuming σ = 1.1, is there
sufficient evidence to support the claim that the mean number of pets owned has changed since 2010 at the α =
0.1 level of significance?
10.6 The Probability of a Type II Error and the Power of the Test
1 Determine the probability of making a Type II error and compute the power of the test.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1) It is desired to test H0 : μ = 40 against H1 : μ < 40 using α = 0.10. The population in question is uniformly
distributed with a standard deviation of 10. A random sample of 36 will be drawn from this population. If μ is
really equal to 35, what is the probability that the hypothesis test would lead the investigator to commit a Type
II error?
A) 0.0427
B) 0.9573
C) 0.4573
D) 0.0854
2) It is desired to test H0 : μ = 12 against H1 : μ ≠ 12 using α = 0.05. The population in question is uniformly
distributed with a standard deviation of 2.0. A random sample of 100 will be drawn from this population. If μ
is really equal to 11.9, what is the value of β associated with this test?
A) 0.0790
B) 0.9210
C) 0.4210
D) 0.0395
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
3) It has been estimated that the G-car obtains a mean of 40 miles per gallon on the highway, and the company
that manufactures the car claims that it exceeds this estimate in highway driving. To support its assertion, the
company randomly selects 36 G-cars and records the mileage obtained for each car over a driving course
similar to that used to obtain the estimate. The following data resulted: x = 41.5 miles per gallon, s = 6 miles per
gallon. Calculate the value of β if the true value of the mean is really 42 miles per gallon. Use α = 0.025.
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MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
4) The probability of making a Type II error is indicated by the letter
A) β
B) α
C) μ
5) A Type II error is an error that
A) fails to reject H0 , given that H1 is true.
D) σ
B) rejects H0 , given that H1 is false.
D) rejects H0 , given that H1 is true.
C) fails to reject H0 , given that H1 is false.
6) In a hypothesis test as the population mean gets closer to the hypothesized mean, β
A) increases.
B) decreases.
C) does not change.
D) becomes the reciprocal of α.
7) The power of the test is
A) 1 - β.
B) α.
C) β.
D) 1 - α.
8) The greater the power of the test the more likely the test will
A) reject H0 when H1 is true.
B) reject H0 when H1 is false.
D) fail to reject H0 when H1 is false.
C) fail to reject H0 when H1 is true.
9) If β is computed to be 0.763, then the power of the test is
A) 0.237.
B) 0.763.
C) 0.263.
10) A power curve is a graphic that plots
A) the power of the test against values of the population mean that make H0 false.
B) the power of the test against values of the population mean that make H0 true.
C) the power of the test against values of the hypothesized mean that make H0 false.
D) the power of the test against values of the hypothesized mean that make H0 true.
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D) 0.737.
Ch. 10 Hypothesis Tests Regarding a Parameter
Answer Key
10.1 The Language of Hypothesis Testing
1 Determine the null and alternative hypotheses.
1) A
2) A
3) A
4) H0 : μ = 3.4, H1 : μ < 3.4
5) H0 : μ = 50.7, H1 : μ ≠ 50.7
6) H0 : μ = $110, H1 : μ > $110
7) H0 : μ = $90, H1 : μ < $90
8) H0 : μ = 4.7, H1 : μ < 4.7
9) H0 : p = 0.5, H1 : p > 0.5
10) H0 : σ = 50, H1 : σ > 50
11) A
12) A
13) A
2 Explain Type I and Type II errors.
1) type I: rejecting H0 : μ = 48.6 when μ = 48.6
type II: failing to reject H0 : μ = 48.6 when μ ≠ 48.6
2) type I: rejecting H0 : μ = $140 when μ ≤ $140
type II: failing to reject H0 : μ = $140 when μ > $140
3) type I: rejecting H0 : μ = $93 when μ ≥ $93
type II: failing to reject H0 : μ = $93 when μ < $93
4) type I: rejecting H0 : p = 0.5 when p ≤ 0.5
type II: failing to reject H0 : p = 0.5 when p > 0.5
5) A
6) A
7) A
8) A
9) A
10) A
11) A
12) A
3 State conclusions to hypothesis tests.
1) A
2) A
3) A
4) A
5) A
6) A
7) A
8) A
9) A
10) A
11) A
12) A
10.2 Hypothesis Tests for a Population Proportion
1 Explain the logic of hypothesis testing.
1) A
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2) A
3) A
4) A
5) A
6) A
2 Test hypotheses about a population proportion.
1) A
2) A
3) A
4) A
5) A
6) A
^
7) The test statistic is z = p - p0
p0 q0
^
p = 8
= 0.114
70
n
The test statistic is z = 0.114 - 0.15
= -0.844
0.15(0.85)
70
8) To determine if the sample size is large enough for the test of hypothesis to work properly, we need to calculate the
interval p0 ± 3σ ^.
p
p0 ± 3σ ^ = p0 ± 3
p
p0 q0
n
= 0.15 ± 3
0.15(0.85)
= 0.15 ± 3(0.040) ⇒ (0.030, 0.270)
80
Since this interval does not include the values of 0 or 1, the sample size of n = 80 is large enough for the test of
hypothesis to work properly.
9) To determine if more than 85% of the firms do not offer any child-care benefits, we test:
H0 : p = 0.85 vs. Ha : p > 0.85
The rejection region requires α = 0.10 in the upper tail of the z distribution. The rejection region is z > z 0.10 = 1.28.
10) At α = 0.10, α < P-value = 0.1112, so H0 cannot be rejected. There is insufficient evidence to indicate that more than
80% of the firms do not offer any child-care benefits.
11) A
12) A
13) A
14) critical value z 0 = -1.28; standardized test statistic ≈ -1.36; reject H0 ; There is sufficient evidence to reject the claim.
15) critical value z 0 = 1.645; standardized test statistic ≈ 1.21; fail to reject H0 ; There is not sufficient evidence to support
the claim p > 0.5. The Republican candidate has no chance.
16) critical value z 0 = -2.33; standardized test statistic ≈ -0.45; fail to reject H0 ; There is not sufficient evidence to support
the airlineʹs claim.
17) critical value z 0 = -1.645; standardized test statistic ≈ -1.33; fail to reject H0 ; There is not sufficient evidence to reject
the claim.
18) critical value z 0 = ±1.96; standardized test statistic ≈ 2.76; reject H0 ; There is sufficient evidence to reject the
universityʹs claim.
19) critical value z 0 = ±1.96; standardized test statistic ≈ 4.43; reject H0 ; There is sufficient evidence to reject the claim that
this is not a biased coin.
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20) (a) The researcher would be testing whether more than 22% of teenagers in the city smoke.
np0 (1 - p0 ) = 17.16 > 10
Critical value z α/2 = 1.645; Test statistic z = 0.72
Do not reject the null hypothesis. There is not sufficient evidence at the α = 0.05 level of significance that more than
22% of teenagers in the city smoke. (b) The researcher would be testing whether more than 24% of teenagers in the city smoke. np0 (1 - p0 ) = 18.24 > 10
Critical value z α/2 = 1.645; Test statistic z = 0.23
Do not reject the null hypothesis. There is not sufficient evidence at the α = 0.05 level of significance that more than
24% of teenagers in the city smoke.
(c) ʺAcceptingʺ the null hypothesis can lead to contradictory conclusions, whereas ʺnot rejectingʺ it does not. If we
accept the null hypothesis, then in part (a) we are ʺacceptingʺ that the population proportion is equal to 0.22 and in
part (b) we are ʺacceptingʺ that the population proportion is equal to 0.24. This means that we have used the same
data to conclude that the population proportion is equal to two different values. If we ʺfail to rejectʺ the null
hypothesis then , rather than concluding that the population proportion is equal to 0.22 (or 0.24) we are simply saying
that these values cannot be ruled out.
3 Test hypotheses about a population proportion using the binomial probability distribution.
1) H0 : p = 0.037 and H1 : p > 0.037
Since 250(0.037)(1 - 0.037) = 8.91 < 10, we calculate the P-value:
P(X ≥ 4) = 1 - P(X < 4) = 1 - P(X ≤ 3) = 1 - 0.016 = 0.984.
Since 0.984 > 0.1, we do not reject H0 .
There is not significant evidence to conclude that the percentage has increased.
2) H0 : p = 0.072 and H1 : p > 0.072
Since 125(0.072)(1 - 0.072) = 8.35 < 10, we calculate the P-value:
P(X ≥ 17) = 1 - P(X < 17) = 1 - P(X ≤ 16) = 1 - 0.992 = 0.008.
Since 0.008 < 0.01, we reject H0 .
There is evidence to conclude that the percentage has increased.
3) H0 : p = 0.059 and H1 : p > 0.059
Since 180(0.059)(1 - 0.059) = 9.99 < 10, we calculate the P-value:
P(X ≥ 12) = 1 - P(X < 12) = 1 - P(X ≤ 11) = 1 - 0.626 = 0.374.
Since 0.374 > 0.05, we do not reject H0 .
There is not significant evidence to conclude that the percentage has increased.
4) A
10.3 Hypothesis Tests for a Population Mean
1 Test hypotheses about a mean.
1) A
2) A
3) A
4) A
5) A
6) A
7) A
8) A
9) A
10) A
11) A
12) A
13) A
14) t0 = ±3.106, standardized test statistic ≈ 1.890, fail to reject H0 ; There is not sufficient evidence to reject the claim.
15) t0 = -1.833, standardized test statistic ≈ -2.189, reject H0 ; There is sufficient evidence to reject the claim
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16) t0 = 1.761, standardized test statistic ≈ 1.452, fail to reject H0 ; There is not sufficient evidence to reject the claim
17) t0 = -1.328, standardized test statistic ≈ -0.894, fail to reject H0 ; There is not sufficient evidence to support the claim
18) t0 = 2.797, standardized test statistic ≈ 1.667, fail to reject H0 ; There is not sufficient evidence to support the claim
19) t0 = ±3.106, standardized test statistic ≈ -0.825, fail to reject H0 ; There is not sufficient evidence to support the claim
20) critical value t0 = -2.539; standardized test statistic ≈ -3.407; reject H0 ; There is sufficient evidence to support the
retailerʹs claim.
21) critical value t0 = ±2.064; standardized test statistic ≈ -1.25; fail to reject H0 ; There is not sufficient evidence to reject
the manufacturerʹs claim.
22) critical value t0 = -1.740; standardized test statistic -2.475; reject H0 ; There is sufficient evidence to support the
shipping firmʹs claim.
23) critical value t0 = -2.539; standardized test statistic ≈ -1.917; fail to reject H0 ; There is not sufficient evidence to
support the government agencyʹs claim.
24) x = 60.21, s = 13.43; critical value t0 = 2.650; standardized test statistic ≈ 0.060; fail to reject H0 ; There is not sufficient
evidence to reject the claim.
25) Standardized test statistic ≈ -1.118; Therefore, at 19 degrees of freedom, the P-value must lie between 0.10 and 0.25.
Since the P-value > α, fail to reject H0 . There is not sufficient evidence to support the bankʹs claim.
26) Standardized test statistic ≈ 1.118; Therefore, at a degree of freedom of 19, the P-value must lie between 0.10 and 0.25.
Since the P-value > α, fail to reject H0 . There is not sufficient evidence to support the storeʹs claim.
27) A
28) A
29) A
2 Understand the difference between statistical significance and practical significance.
1) A
2) A
10.4 Hypothesis Tests for a Population Standard Deviation
1 Test hypotheses about a population standard deviation.
1) A
2) A
3) A
4) A
5) A
6) A
7) A
8) A
9) A
2
2
10) critical values χ L = 3.816 and χ R = 21.920; standardized test statistic χ 2 = 9.209; fail to reject H0 ; There is not
sufficient evidence to reject the claim.
2
11) critical value χ 0 = 6.571; standardized test statistic χ 2 ≈ 11.667; fail to reject H0 ; There is not sufficient evidence to
reject the claim.
2
12) critical value χ 0 = 36.191; standardized test statistic χ 2 ≈ 36.813; reject H0 ; There is sufficient evidence to reject the
claim.
2
13) critical value χ 0 = 33.409; standardized test statistic χ 2 ≈ 24.158; fail to reject H0 ; There is not sufficient evidence to
reject the claim.
2
14) critical value χ 0 = 18.114; standardized test statistic χ 2 ≈ 16.875; reject H0 ; There is sufficient evidence to support the
claim.
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2
2
15) critical values χ L = 1.735 and χ R = 23.589; standardized test statistic χ 2 ≈ 9.926; fail to reject H0 ; There is not
sufficient evidence to support the claim.
2
2
16) critical values χ L = 3.816 and χ R = 21.920; standardized test statistic χ 2 ≈ 9.267; fail to reject H0 ; There is not
sufficient evidence to reject the claim.
2
17) critical value χ 0 = 6.571; standardized test statistic χ 2 ≈ 11.605; fail to reject H0 ; There is not sufficient evidence to
reject the claim.
2
18) critical value χ 0 = 36.191; standardized test statistic χ 2 ≈ 36.766; reject H0 ; There is sufficient evidence to reject the
claim.
2
19) critical value χ 0 = 33.409; standardized test statistic χ 2 ≈ 24.009; fail to reject H0 ; There is not sufficient evidence to
support the claim.
2
20) critical value χ 0 = 18.114; standardized test statistic χ 2 ≈ 16.809; reject H0 ; There is sufficient evidence to support the
claim.
2
2
21) critical values χ L = 1.735 and χ R = 23.589; standardized test statistic χ 2 ≈ 9.919; fail to reject H0 ; There is not
sufficient evidence to support the claim.
2
2
22) critical values χ L = 3.565 and χ R = 29.819; standardized test statistic χ 2 ≈ 10.42; fail to reject H0 ; There is not
sufficient evidence to reject the claim.
2
23) critical value χ 0 = 10.117; standardized test statistic χ 2 ≈ 9.098; reject H0 ; There is sufficient evidence to support the
claim.
2
24) critical value χ 0 = 124.342; standardized test statistic χ 2 = 144; reject H0 ; There is sufficient evidence to support the
claim.
2
2
25) critical values χ L = 35.534 and χ R = 91.952; standardized test statistic χ 2 ≈ 78.634; fail to reject H0 ; There is not
sufficient evidence to reject the claim.
2
26) critical value χ 0 = 90.531; standardized test statistics χ 2 = 149.527; reject H0 ; There is sufficient evidence to support
the claim.
2
27) critical value χ 0 = 63.691; standardized test statistic χ 2 = 53.824; fail to reject H0 ; There is not sufficient evidence to
support the claim.
28) Since α = 0.01 > p = 0.0085, H0 can be rejected. There is sufficient evidence to indicate that the variance in the amount
of serum injected exceeds 0.08.
10.5 Putting It Together: Which Method Do I Use?
1 Determine the appropriate hypothesis test to perform.
1) H0 : p = 0.36, H1 : p ≠ 0.36, Z = 1.15, P-value = 0.2485; do not reject H0 . There is not sufficient evidence at the α = 0.1
level of significance to support the practitionerʹs claim that the percentage of adults who are morbidly obese has
change since 2010.
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2) H0 : μ = $806, H1 :μ < $806, T0 = -3.61, P-value = 0.0005; reject H0 . There is sufficient evidence at the α = 0.05 level of
significance to support the salespersonʹs claim that the mean expenditure for auto insurance is less than the 2010
amount.
3) H0 : σ = 2 mg, H1 :σ > 2 mg, χ 2 = 41.76, P-value = 0.0590; do not to reject H0 . There is not sufficient evidence at the α =
0.05 level of significance to support the claim that the standard deviation of the amount of the active ingredient is
greater than 2 mg.
4) H0 : μ = 1.9, H1 :μ ≠ 1.9, Z0 = -2.91, P-value = 0.0036; reject H0 . There is sufficient evidence at the α = 0.10 level of
significance to support the claim that the mean number of pets owned per household has changed since 2010.
10.6 The Probability of a Type II Error and the Power of the Test
1 Determine the probability of making a Type II error and compute the power of the test.
1) A
2) A
3) Since the alternative hypothesis is H1 : μ > 40, the test is one-tailed. Thus, α = 0.025 is required in the upper tail of the
z distribution, and we have z 0.025 = 1.96. The value of x on the border between the rejection region and the
acceptance region is found using
z = σ
x - 40
⇒ x = z
+ 40 ⇒ x = 1.96
n
σ/ n
β = P(x < 41.96, when μa = 42) = P z < 6
+ 40 ⇒ x = 41.96
36
41.96 - 42
= P(z < -0.04) = 0.5 - 0.0160 = 0.4840
6/ 36
4) A
5) A
6) A
7) A
8) A
9) A
10) A
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