Ch. 10 Hypothesis Tests Regarding a Parameter

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 0 : = 9.5

H 1 : 9.5

A) Two-tailed,

B) Two-tailed, x

C) Right-tailed,

D) Left-tailed, x

2) H 0 : p = 0.86 H 1 : p > 0.86

A) Right-tailed, p

B) Left-tailed, p

C)

Right-tailed,

^ p

D)

Left-tailed,

^ p

3) H 0 : = 8.5 H 1 : < 8.5 A) Left-tailed,

B) Right-tailed,

C) Right-tailed,

D) Left-tailed, s

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 employees IRA is at most 3.9 percent. Write the null and alternative hypotheses.

5) The mean age of lawyers in New York is 54.5 years. Write the null and alternative hypotheses.

6) The mean repair bill of cars is greater than $150. Write the null and alternative hypotheses.

7) The mean utility bill in one city during the summer was less than $95. Write the null and alternative hypotheses.

8) The mean annual return for an employees IRA is at most 3.7 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 = sign.

A) null

B) alternative

C) explanatory

D) conditional

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13) A hypothesis test is a two-tailed 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. 14) The mean age of judges in Los Angeles is 51.1 years. Identify the type I and type II errors for the hypothesis test of this claim.

15) The mean cost of textbooks for one class is greater than $130. Identify the type I and type II errors for the hypothesis test of this claim.

16) The mean monthly cell phone bill for one household was less than $ 99. Identify the type I and type II errors for the hypothesis test of this claim.

17) 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.

18) 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 17 new rackets at 56 psi. Upon receiving the rackets, the customer measured the tension of each and calculated the following summary statistics: x = 55 psi, s = 3.5 psi. In order to conduct the test, the customer selected a significance level of = .01. Interpret this value. A) The probability of concluding that the true mean is less than 56 psi when in fact it is equal to 56 psi is only .01. B) The smallest value of that you can use and still reject H0 is 0.01. C) The probability of making a Type II error is 0.99. D) There is a 1% chance that the sample will be biased.

19) True or False: If I specify to be equal to 0.18, then the value of must be 0.82.

A) False

B) True

20) What is the probability associated with not making a Type II error?

A) (1 - )

B)

C)

D) (1 - )

21) We never conclude Accept H0 in a test of hypothesis. This is because:

A) = p(Type II error) is not known.

B) is the probability of a Type I error.

C) The rejection region is not known.

D) The p-value is not small enough.

22) 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 error

23) 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 error

24) The level of significance, , is the probability of making a

A) Type I error

B) Type II error

C) Correct decision

D) Type error

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25) True or False: Type I and Type II errors are independent events.

A) False

B) True

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. 26) The mean age of principals in a local school district is 58.7 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 = 58.7. B) There is not sufficient evidence to reject the claim = 58.7. C) There is sufficient evidence to support the claim = 58.7. D) There is not sufficient evidence to support the claim = 58.7.

27) The mean age of professors at a university is 52.2 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 reject the claim = 52.2. B) There is sufficient evidence to reject the claim = 52.2. C) There is sufficient evidence to support the claim = 52.2. D) There is not sufficient evidence to support the claim = 52.2.

28) The mean age of professors at a university is greater than 51.2 years. If a hypothesis test is performed, how

should you interpret a decision that rejects the null hypothesis? A) There is sufficient evidence to support the claim > 51.2. B) There is sufficient evidence to reject the claim > 51.2. C) There is not sufficient evidence to reject the claim > 51.2. D) There is not sufficient evidence to support the claim > 51.2.

29) The mean age of judges in Dallas is greater than 58.8 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 > 58.8. B) There is sufficient evidence to reject the claim > 58.8. C) There is not sufficient evidence to reject the claim > 58.8. D) There is sufficient evidence to support the claim > 58.8.

30) The mean monthly gasoline bill for one household is greater than $120. 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 > $120. B) There is sufficient evidence to reject the claim > $120. C) There is not sufficient evidence to reject the claim > $120. D) There is not sufficient evidence to support the claim > $120.

31) The mean monthly gasoline bill for one household is greater than $120. 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 > $120. B) There is sufficient evidence to reject the claim > $120. C) There is not sufficient evidence to reject the claim > $120. D) There is sufficient evidence to support the claim > $120.

Page 3

32) The mean number of rushing yards for one NFL team was less than 105 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 < 105. B) There is sufficient evidence to reject the claim < 105. C) There is not sufficient evidence to reject the claim < 105. D) There is not sufficient evidence to support the claim < 105.

33) The mean number of rushing yards for one NFL team was less than 110 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 < 110. B) There is sufficient evidence to reject the claim < 110. C) There is not sufficient evidence to reject the claim < 110. D) There is sufficient evidence to support the claim < 110.

34) The dean of a major university claims that the mean number of hours students study at her University (per

day) is at most 3.2 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 3.2. B) There is not sufficient evidence to reject the claim 3.2. C) There is sufficient evidence to support the claim 3.2. D) There is not sufficient evidence to support the claim 3.2.

35) The dean of a major university claims that the mean number of hours students study at her University (per

day) is at most 3.8 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 3.8. B) There is sufficient evidence to reject the claim 3.8. C) There is sufficient evidence to support the claim 3.8. D) There is not sufficient evidence to support the claim 3.8.

36) 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.

37) 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 right-tailed test with = 0.05.

A) 1.645

B) 1.96

C) 1.28

D) 2.33

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2) Find the critical value for a two-tailed test with = 0.10.

A) ?1.645

B) ?1.28

C) ?2.575

D) ?1.96

3) Suppose you want to test the claim that = 3.5. Given a sample size of n = 51 and a level of significance of = 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 1.645 or less than -1.645 D) Reject H0 if the standardized test statistic is greater than 1.96 or less than -1.96

4) Suppose you want to test the claim that > 25.6. Given a sample size of n = 43 and a level of significance of = 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 1.28. C) Reject H0 if the standardized test statistic is greater than 2.575. D) Reject H0 if the standardized test statistic is greater than 1.96.

5) Suppose you want to test the claim that < 65.4. Given a sample size of n = 35 and a level of significance of = 0.10, when should you reject H0? A) Reject H0 if the standardized test statistic is less than -1.28. B) Reject H0 if the standardized test is less than -1.645. C) Reject H0 if the standardized test statistic is less than -2.33. D) Reject H0 if the standardized test statistic is less than -2.575.

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. (Classical Approach)

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Provide an appropriate response.

7) The business college computing center wants to determine the proportion of business students who have

personal computers (PCs) at home. If the proportion exceeds 25%, then the lab will scale back a proposed

enlargement of its facilities. Suppose 200 business students were randomly sampled and 65 have PCs at home.

Find the rejection region for this test using = 0.01.

A) Reject H0 if z > 2.33.

B) Reject H0 if z < -2.33.

C) Reject H0 if z > 2.575 or z < -2.575.

D) Reject H0 if z = 2.33.

8) The business college computing center wants to determine the proportion of business students who have personal computers (PCs) at home. If the proportion exceeds 25%, then the lab will scale back a proposed enlargement of its facilities. Suppose 200 business students were randomly sampled and 65 have PCs 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.

Page 5

9) 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

10) 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 = -1.95. Can we conclude that H0

should be rejected at the a) =0.10, b) = 0.05, and c) = 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

11) A nationwide survey claimed that at least 65% 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 = 0.5? A) You would need 57 or less parents to support spanking to refute the claim. B) You would need exactly 57 parents to support spanking to refute the claim. C) You would need 58 or less parents to support spanking to refute the claim. D) You would need more than 57 parents to support spanking to refute the claim.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

12) 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. Calculate the test statistic used by the researchers for this test of hypothesis. Round to the nearest thousandth.

13) A method currently used by doctors to screen women for possible breast cancer fails to detect cancer in 20% 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 90 women known to have breast cancer were screened using the new method. Of these, the new method failed to detect cancer in 10. Is the sample size sufficiently large in order to conduct this test of hypothesis? Explain. Round to the nearest thousandth.

14) Increasing numbers of businesses are offering child-care benefits for their workers. However, one union claims that more than 90% of firms in the manufacturing sector still do not offer any child-care benefits to their workers. A random sample of 260 manufacturing firms is selected, and only 34 of them offer child-care benefits. Specify the rejection region that the union will use when testing at = 0.10.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

15) 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

16) Determine the critical value, z0, to test the claim about the population proportion p 0.325 given n = 42 and

^ p

=

0.247.

Use

=

0.05.

A) ?1.96

B) ?2.575

C) ?1.645

D) ?2.33

17) Determine the standardized test statistic, z, to test the claim about the population proportion p 0.700 given

n

=

50

and

^ p

= 0.612. Use = 0.10.

A) -1.36

B) -1.28

C) -2.18

D) -3.01

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SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

18)

Test

the

claim

about

the

population

proportion

p

<

0.850

given

n

=

60

and

^ p

=

0.656.

Use

=

0.05.

19) 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 = 0.05.

20) 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

=

0.01,

test

the

airlines

claim.

Round

^ p

to

the

nearest

thousandth

when

calculating the test statistic.

21) 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 = 0.05, test the claim.

22) 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

=

0.05.

Round

^ p

to

the

nearest ten-thousandth when calculating the test statistic.

23) A coin is tossed 1000 times and 570 heads appear. At = 0.05, test the claim that this is not a biased coin.

24) In one city, 25 out of 100 randomly sampled teenagers say that they smoke. (a) Consider the hypotheses H0: p = 0.2 versus H1: p > 0.2. Explain what the researcher would be testing. Perform the test at the = 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

accepting the statement in the null hypothesis versus not rejecting the statement in the null hypothesis.

3 Test hypotheses about a population proportion. (P-Value Approach)

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Provide an appropriate response.

25) The business college computing center wants to determine the proportion of business students who have

personal computers (PCs) at home. If the proportion differs from 30%, then the lab will modify a proposed enlargement of its facilities. Suppose a hypothesis test is conducted and the test statistic is 2.5. Find the P-value for a two-tailed test of hypothesis.

A) 0.0124

B) 0.0062

C) 0.4876

D) 0.4938

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

26) 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 490 manufacturing firms is selected and asked if they offer child -care benefits. Suppose the P-value for this test was reported to be p = 0.1070. State the conclusion of interest to the union. Use = 0.10.

27)

Test

the

claim

about

the

population

proportion

p

<

0.850

given

n

=

60

and

^ p

=

0.656.

Use

=

0.05.

28) 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 = 0.05.

Page 7

29) 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

=

0.01,

test

the

airlines

claim.

Round

^ p

to

the

nearest

thousandth

when

calculating the test statistic.

30) 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 = 0.05, test the claim.

31) 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

=

0.05.

Round

^ p

to

the

nearest ten-thousandth when calculating the test statistic.

32) A coin is tossed 1000 times and 540 heads appear. At = 0.05, test the claim that this is not a biased coin.

4 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. 33) 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 researchers claim at the = 0.1 level of significance.

34) 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 researchers claim at the = 0.01 level of significance.

35) 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 dealers claim at the = 0.05 level of significance.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

36) 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. (Classical Approach)

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 = 0.005 level of significance

with 28 degrees of freedom.

A) 2.763

B) 1.701

C) -2.763

D) 2.771

2) Determine the critical value for a left-tailed test of a population mean at the = 0.025 level of significance

based on a sample size of n = 18.

A) -2.11

B) -3.222

C) 2.110

D) 2.101

3) Determine the critical values for a two-tailed test of a population mean at the = 0.01 level of significance

based on a sample size of n = 21.

A) ?2.845

B) ?2.831

C) ?2.528

D) ?2.518

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