doane - stats answer key chap 008

Chapter 08Sampling Distributions and Estimation

True / False Questions1.The expected value of an unbiased estimator is equal to the parameter whose value is being estimated.TrueFalse

2.All estimators are biased since sampling errors always exist to some extent.TrueFalse

3.An estimator must be unbiased if you are to use it for statistical analysis.TrueFalse

4.The efficiency of an estimator depends on the variance of the estimator's sampling distribution.TrueFalse

5.In comparing estimators, the more efficient estimator will have a smaller standard error.TrueFalse

6.A 90 percent confidence interval will be wider than a 95 percent confidence interval, ceteris paribus.TrueFalse

7.In constructing a confidence interval for the mean, the z distribution provides a result nearly identical to the t distribution when n is large.TrueFalse

8.The Central Limit Theorem says that, if n exceeds 30, the population will be normal.TrueFalse

9.The Central Limit Theorem says that a histogram of the sample means will have a bell shape, even if the population is skewed and the sample is small.TrueFalse

10.The confidence level refers to the procedure used to construct the confidence interval, rather than to the particular confidence interval we have constructed.TrueFalse

11.The Central Limit Theorem guarantees an approximately normal sampling distribution when n is sufficiently large.TrueFalse

12.A sample of size 5 shows a mean of 45.2 and a sample standard deviation of 6.4. The standard error of the sample mean is approximately 2.86.TrueFalse

13.As n increases, the width of the confidence interval will decrease, ceteris paribus.TrueFalse

14.As n increases, the standard error decreases.TrueFalse

15.A higher confidence level leads to a narrower confidence interval, ceteris paribus.TrueFalse

16.When the sample standard deviation is used to construct a confidence interval for the mean, we would use the Student's t distribution instead of the normal distribution.TrueFalse

17.As long as the sample is more than one item, the standard error of the sample mean will be smaller than the standard deviation of the population.TrueFalse

18.For a sample size of 20, a 95 percent confidence interval using the t distribution would be wider than one constructed using the z distribution.TrueFalse

19.In constructing a confidence interval for a mean, the width of the interval is dependent on the sample size, the confidence level, and the population standard deviation.TrueFalse

20.In constructing confidence intervals, it is conservative to use the z distribution when n 30.TrueFalse

21.The Central Limit Theorem can be applied to the sample proportion.TrueFalse

22.The distribution of the sample proportion p = x/n is normal when n 30.TrueFalse

23.The standard deviation of the sample proportion p = x/n increases as n increases.TrueFalse

24.A 95 percent confidence interval constructed around p will be wider than a 90 percent confidence interval.TrueFalse

25.The sample proportion is always the midpoint of a confidence interval for the population proportion.TrueFalse

26.The standard error of the sample proportion is largest when = .50.TrueFalse

27.The standard error of the sample proportion does not depend on the confidence level.TrueFalse

28.To narrow the confidence interval for , we can either increase n or decrease the level of confidence.TrueFalse

29.Ceteris paribus, the narrowest confidence interval for is achieved when p = .50.TrueFalse

30.The statistic p = x/n may be assumed normally distributed when np 10 and n(1 - p) 10.TrueFalse

31.The Student's t distribution is always symmetric and bell-shaped, but its tails lie above the normal.TrueFalse

32.The confidence interval half-width when = .50 is called the margin of error.TrueFalse

33.Based on the Rule of Three, if no events occur in n independent trials we can set the upper 95 percent confidence bound at 3/n.TrueFalse

34.The sample standard deviation s is halfway between the lower and upper confidence limits for the population (i.e., the confidence interval is symmetric around s).TrueFalse

35.In a sample size calculation, if the confidence level decreases, the size of the sample needed will increase.TrueFalse

36.To calculate the sample size needed for a survey to estimate a proportion, the population standard deviation must be known.TrueFalse

37.Assuming that = .50 is a quick and conservative approach to use in a sample size calculation for a proportion.TrueFalse

38.To estimate the required sample size for a proportion, one method is to take a small pilot sample to estimate and then apply the sample size formula.TrueFalse

39.To estimate , you typically need a sample size equal to at least 5 percent of your population.TrueFalse

40.To estimate a proportion with a 4 percent margin of error and a 95 percent confidence level, the required sample size is over 800.TrueFalse

41.Approximately 95 percent of the population X values will lie within the 95 percent confidence interval for the mean.TrueFalse

42.A 99 percent confidence interval has more confidence but less precision than a 95 percent confidence interval.TrueFalse

43.Sampling variation is not controllable by the statistician.TrueFalse

44.The sample mean is not a random variable when the population parameters are known.TrueFalse

45.The finite population correction factor (FPCF) can be ignored if n = 7 and N = 700.TrueFalse

46.In constructing a confidence interval, the finite population correction factor (FPCF) can be ignored if samples of 12 items are drawn from a population of 300 items.TrueFalse

47.The finite population correction factor (FPCF) can be ignored when the sample size is large relative to the population size.TrueFalse

Multiple Choice Questions48.A sampling distribution describes the distribution of:

A.a parameter.

B.a statistic.

C.either a parameter or a statistic.

D.neither a parameter nor a statistic.

49.As the sample size increases, the standard error of the mean:

A.increases.

B.decreases.

C.may increase or decrease.

50.Which statement is most nearly correct, other things being equal?

A.Doubling the sample size will cut the standard error of the mean in half.

B.The standard error of the mean depends on the population size.

C.Quadrupling the sample size roughly halves the standard error of the mean.

D.The standard error of the mean depends on the confidence level.

51.The width of a confidence interval for is not affected by:

A.the sample size.

B.the confidence level.

C.the standard deviation.

D.the sample mean.

52.The Central Limit Theorem (CLT) implies that:

A.the population will be approximately normal if n 30.

B.repeated samples must be taken to obtain normality.

C.the distribution of the mean is approximately normal for large n.

D.the mean follows the same distribution as the population.

53.The owner of Limp Pines Resort wanted to know the average age of its clients. A random sample of 25 tourists is taken. It shows a mean age of 46 years with a standard deviation of 5 years. The width of a 98 percent CI for the true mean client age is approximately:

A. 1.711 years.

B. 2.326 years.

C. 2.492 years.

D. 2.797 years.

54.In constructing a confidence interval for a mean with unknown variance with a sample of 25 items, Bob used z instead of t. "Well, at least my interval will be wider than necessary, so it was a conservative error," said he. Is Bob's statement correct?

A.Yes.

B.No.

C.It depends on .

55.A random sample of 16 ATM transactions at the Last National Bank of Flat Rock revealed a mean transaction time of 2.8 minutes with a standard deviation of 1.2 minutes. The width (in minutes) of the 95 percent confidence interval for the true mean transaction time is:

A. 0.639

B. 0.588

C. 0.300

D. 2.131

56.We could narrow a 95 percent confidence interval by:

A.using 99 percent confidence.

B.using a larger sample.

C.raising the standard error.

57.The owner of Torpid Oaks B&B wanted to know the average distance its guests had traveled. A random sample of 16 guests showed a mean distance of 85 miles with a standard deviation of 32 miles. The 90 percent confidence interval (in miles) for the mean is approximately:

A.(71.0, 99.0)

B.(71.8, 98.2)

C.(74.3, 95.7)

D.(68.7, 103.2)

58.A highway inspector needs an estimate of the mean weight of trucks crossing a bridge on the interstate highway system. She selects a random sample of 49 trucks and finds a mean of 15.8 tons with a sample standard deviation of 3.85 tons. The 90 percent confidence interval for the population mean is:

A.14.72 to 16.88 tons.

B.14.90 to 16.70 tons.

C.14.69 to 16.91 tons.

D.14.88 to 16.72 tons.

59.To determine a 72 percent level of confidence for a proportion, the value of z is approximately:

A. 1.65

B. 0.77

C. 1.08

D. 1.55

60.To estimate the average annual expenses of students on books and class materials a sample of size 36 is taken. The sample mean is $850 and the sample standard deviation is $54. A 99 percent confidence interval for the population mean is:

A.$823.72 to $876.28

B.$832.36 to $867.64

C.$826.82 to $873.18

D.$825.48 to $874.52

61.In constructing a 95 percent confidence interval, if you increase n to 4n, the width of your confidence interval will (assuming other things remain the same) be:

A.about 25 percent of its former width.

B.about two times wider.

C.about 50 percent of its former width.

D.about four times wider.

62.Which of the following is not a characteristic of the t distribution?

A.It is a continuous distribution.

B.It has a mean of 0.

C.It is a symmetric distribution.

D.It approaches z as degrees of freedom decrease.

63.Which statement is incorrect? Explain.

A.If p = .50 and n = 100, the standard error of the sample proportion is .05.

B.In a sample size calculation for estimating , it is conservative to assume = .50.

C.If n = 250 and p = .06, we cannot assume normality in a confidence interval for .

64.What is the approximate width of a 90 percent confidence interval for the true population proportion if there are 12 successes in a sample of 25?

A. .196

B. .164

C. .480

D. .206

65.A poll showed that 48 out of 120 randomly chosen graduates of California medical schools last year intended to specialize in family practice. What is the width of a 90 percent confidence interval for the proportion that plan to specialize in family practice?

A. .0447

B. .0736

C. .0876

D. .0894

66.What is the approximate width of an 80 percent confidence interval for the true population proportion if there are 12 successes in a sample of 80?

A. .078

B. .066

C. .051

D. .094

67.A random sample of 160 commercial customers of PayMor Lumber revealed that 32 had paid their accounts within a month of billing. The 95 percent confidence interval for the true proportion of customers who pay within a month would be:

A.0.148 to 0.252

B.0.138 to 0.262

C.0.144 to 0.256

D.0.153 to 0.247

68.A random sample of 160 commercial customers of PayMor Lumber revealed that 32 had paid their accounts within a month of billing. Can normality be assumed for the sample proportion?

A.Yes.

B.No.

C.Need more information to say.

69.The conservative sample size required for a 95 percent confidence interval for with an error of 0.04 is:

A.271.

B.423.

C.385.

D.601.

70.Last week, 108 cars received parking violations in the main university parking lot. Of these, 27 had unpaid parking tickets from a previous violation. Assuming that last week was a random sample of all parking violators, find the 95 percent confidence interval for the percentage of parking violators that have prior unpaid parking tickets.

A.18.1 to 31.9 percent.

B.16.8 to 33.2 percent.

C.15.3 to 34.7 percent.

D.19.5 to 30.5 percent.

71.In a random sample of 810 women employees, it is found that 81 would prefer working for a female boss. The width of the 95 percent confidence interval for the proportion of women who prefer a female boss is:

A. .0288

B. .0105

C. .0207

D. .0196

72.Jolly Blue Giant Health Insurance (JBGHI) is concerned about rising lab test costs and would like to know what proportion of the positive lab tests for prostate cancer are actually proven correct through subsequent biopsy. JBGHI demands a sample large enough to ensure an error of 2 percent with 90 percent confidence. What is the necessary sample size?

A.4,148

B.2,401

C.1,692

D.1,604

73.A university wants to estimate the average distance that commuter students travel to get to class with an error of 3 miles and 90 percent confidence. What sample size would be needed, assuming that travel distances are normally distributed with a range of X = 0 to X = 50 miles, using the Empirical Rule 3 to estimate .

A.About 28 students

B.About 47 students

C.About 30 students

D.About 21 students

74.A financial institution wishes to estimate the mean balances owed by its credit card customers. The population standard deviation is $300. If a 99 percent confidence interval is used and an interval of $75 is desired, how many cardholders should be sampled?

A.3382

B.629

C.87

D.107

75.A company wants to estimate the time its trucks take to drive from city A to city B. The standard deviation is known to be 12 minutes. What sample size is required in order that error will not exceed 2 minutes, with 95 percent confidence?

A.12 observations

B.139 observations

C.36 observations

D.129 observations

76.In a large lecture class, the professor announced that the scores on a recent exam were normally distributed with a range from 51 to 87. Using the Empirical Rule 3 to estimate , how many students would you need to sample to estimate the true mean score for the class with 90 percent confidence and an error of 2?

A.About 17 students

B.About 35 students

C.About 188 students

D.About 25 students

77.Using the conventional polling definition, find the margin of error for a customer satisfaction survey of 225 customers who have recently dined at Applebee's.

A. 5.0 percent

B. 4.2 percent

C. 7.1 percent

D. 6.5 percent

78.A marketing firm is asked to estimate the percent of existing customers who would purchase a "digital upgrade" to their basic cable TV service. The firm wants 99 percent confidence and an error of 5 percent. What is the required sample size (to the next higher integer)?

A.664

B.625

C.801

D.957

79.An airport traffic analyst wants to estimate the proportion of daily takeoffs by small business jets (as opposed to commercial passenger jets or other aircraft) with an error of 4 percent with 90 percent confidence. What sample size should the analyst use?

A.385

B.601

C.410

D.423

80.Ersatz Beneficial Insurance wants to estimate the cost of damage to cars due to accidents. The standard deviation of the cost is known to be $200. They want to estimate the mean cost using a 95 percent confidence interval within $10. What is the minimum sample size n?

A.1083

B.4002

C.1537

D.2301

81.Professor York randomly surveyed 240 students at Oxnard University and found that 150 of the students surveyed watch more than 10 hours of television weekly. Develop a 95 percent confidence interval to estimate the true proportion of students who watch more than 10 hours of television each week. The confidence interval is:

A..533 to .717

B..564 to .686

C..552 to .698

D..551 to .739

82.Professor York randomly surveyed 240 students at Oxnard University and found that 150 of the students surveyed watch more than 10 hours of television weekly. How many additional students would Professor York have to sample to estimate the proportion of all Oxnard University students who watch more than 10 hours of television each week within 3 percent with 99 percent confidence?

A.761

B.1001

C.1489

D.1728

83.The sample proportion is in the middle of the confidence interval for the population proportion:

A.in any sample.

B.only if the samples are large.

C.only if is not too far from .50.

84.For a sample of size 16, the critical values of chi-square for a 95 percent confidence interval for the population variance are:

A.6.262, 27.49

B.6.908, 28.85

C.5.629, 26.12

D.7.261, 25.00


A.6.262, 27.49

B.6.908, 28.85

C.3.940, 18.31

D.3.247, 20.48


A.6.262, 27.49

B.5.697, 35.72

C.5.629, 26.12

D.7.261, 25.00

87.Which of the following statements is most nearly correct, other things being equal?

A.Using Student's t instead of z makes a confidence interval narrower.

B.The table values of z and t are about the same when the mean is large.

C.For a given confidence level, the z value is always smaller then the t value.

D.Student's t is rarely used because it is more conservative to use z.

88.The Central Limit Theorem (CLT):

A.applies only to samples from normal populations.

B.applies to any population.

C.applies best to populations that are skewed.

D.applies only when and are known.

89.In which situation may the sample proportion safely be assumed to follow a normal distribution?

A.12 successes in a sample of 72 items

B.8 successes in a sample of 40 items

C.6 successes in a sample of 200 items

D.4 successes in a sample of 500 items


A.n = 100, = .06

B.n = 250, = .02

C.n = 30, = .50

D.n = 500, = .01

91.If = 12, find the sample size to estimate the mean with an error of 4 and 95 percent confidence (rounded to the next higher integer).

A.75

B.35

C.58

D.113


A.426

B.512

C.267

D.188

93.Sampling error can be avoided:

A.by using an unbiased estimator.

B.by eliminating nonresponses (e.g., older people).

C.by no method under the statistician's control.

D.either by using an unbiased estimator or by eliminating nonresponse.

94.A consistent estimator for the mean:

A.converges on the true parameter as the variance increases.

B.converges on the true parameter as the sample size increases.

C.consistently follows a normal distribution.

D.is impossible to obtain using real sample data.

95.Concerning confidence intervals, which statement is most nearly correct?

A.We should use z instead of t when n is large.

B.We use the Student's t distribution when is unknown.

C.We use the Student's t distribution to narrow the confidence interval.

96.The standard error of the mean decreases when:

A.the sample size decreases.

B.the standard deviation increases.

C.the standard deviation decreases or n increases.

D.the population size decreases.

97.For a given sample size, the higher the confidence level, the:

A.more accurate the point estimate.

B.smaller the standard error.

C.smaller the interval width.

D.greater the interval width.

98.A sample is taken and a confidence interval is constructed for the mean of the distribution. At the center of the interval is always which value?

A.The sample mean

B.The population mean

C.Neither nor since with a sample anything can happen

D.Both and as long as there are not too many outliers

99.If a normal population has parameters = 40 and = 8, then for a sample size n = 4:

A.the standard error of the sample mean is approximately 2.

B.the standard error of the sample mean is approximately 4.

C.the standard error of the sample mean is approximately 8.

D.the standard error of the sample mean is approximately 10.

Short Answer Questions100.On the basis of a survey of 545 television viewers, a statistician has constructed a confidence interval and estimated that the proportion of people who watched the season premiere of Glee is between .16 and .24. What level of confidence did the statistician use in constructing this interval? Explain carefully, showing all steps in your reasoning.

101.Read the news story below. Using the 95 percent confidence level, what sample size would be needed to estimate the true proportion of stores selling cigarettes to minors with an error of 3 percent? Explain carefully, showing all steps in your reasoning.

102.In a survey, 858 out of 2600 homeowners said they expected good economic conditions to continue for the next 12 months. Construct a 95 percent confidence interval for "good times" in the next 12 months.

103.Fulsome University has 16,059 students. In a sample of 200 students, 12 were born outside the United States. Construct a 95 percent confidence interval for the true population proportion. How large a sample is needed to estimate the true proportion of Fulsome students who were born outside the United States with an error of 2.5 percent and 95 percent confidence? Show your work and explain fully.

104.List differences and similarities between Student's t and the standard normal distribution.

105.Why does pose a problem for sample size calculation for a mean? How can be approximated when it is unknown?

Chapter 08 Sampling Distributions and Estimation Answer Key

True / False Questions1.The expected value of an unbiased estimator is equal to the parameter whose value is being estimated.TRUEAn unbiased estimator's expected value is the true parameter value.

AACSB: AnalyticBlooms: RememberDifficulty: 1 EasyLearning Objective: 08-02 Explain the desirable properties of estimators.Topic: Estimators and Sampling Error

2.All estimators are biased since sampling errors always exist to some extent.FALSESome estimators are systematically biased, regardless of sampling error.


3.An estimator must be unbiased if you are to use it for statistical analysis.FALSEAn estimator can be useful as long as its bias is known.

AACSB: AnalyticBlooms: RememberDifficulty: 2 MediumLearning Objective: 08-02 Explain the desirable properties of estimators.Topic: Estimators and Sampling Error

4.The efficiency of an estimator depends on the variance of the estimator's sampling distribution.TRUEEfficiency is measured by the variance of the estimator's sampling distribution.


5.In comparing estimators, the more efficient estimator will have a smaller standard error.TRUEEfficiency is measured by the variance of the estimator's sampling distribution.


6.A 90 percent confidence interval will be wider than a 95 percent confidence interval, ceteris paribus.FALSEWe can make a more precise statement about the true parameter if we are willing to sacrifice some confidence. For example, z.025 = 1.960 (for 95 percent confidence) gives a wider interval than z.05 = 1.645 (for 90 percent confidence). The proffered statement would also hold true for the Student's t distribution.

AACSB: AnalyticBlooms: UnderstandDifficulty: 1 EasyLearning Objective: 08-05 Construct a 90; 95; or 99 percent confidence interval for .Topic: Confidence Interval for a Mean () with Known

7.In constructing a confidence interval for the mean, the z distribution provides a result nearly identical to the t distribution when n is large.TRUEStudent's t approaches z as sample size increases.

AACSB: AnalyticBlooms: RememberDifficulty: 1 EasyLearning Objective: 08-06 Know when to use Student's t instead of z to estimate .Topic: Confidence Interval for a Mean () with Unknown

8.The Central Limit Theorem says that, if n exceeds 30, the population will be normal.FALSEThe population cannot be changed.

AACSB: AnalyticBlooms: RememberDifficulty: 2 MediumLearning Objective: 08-03 State the Central Limit Theorem for a mean.Topic: Sample Mean and the Central Limit Theorem

9.The Central Limit Theorem says that a histogram of the sample means will have a bell shape, even if the population is skewed and the sample is small.FALSEA large sample size may be required if the population is skewed.


10.The confidence level refers to the procedure used to construct the confidence interval, rather than to the particular confidence interval we have constructed.TRUEA particular interval either does or does not contain the true parameter.

AACSB: AnalyticBlooms: UnderstandDifficulty: 2 MediumLearning Objective: 08-05 Construct a 90; 95; or 99 percent confidence interval for .Topic: Confidence Interval for a Mean () with Known

11.The Central Limit Theorem guarantees an approximately normal sampling distribution when n is sufficiently large.TRUEYes, although a large sample size may be required if the population is skewed.

AACSB: AnalyticBlooms: UnderstandDifficulty: 2 MediumLearning Objective: 08-03 State the Central Limit Theorem for a mean.Topic: Sample Mean and the Central Limit Theorem

12.A sample of size 5 shows a mean of 45.2 and a sample standard deviation of 6.4. The standard error of the sample mean is approximately 2.86.TRUEThe standard error is the standard deviation divided by the square root of the sample size.

AACSB: AnalyticBlooms: ApplyDifficulty: 1 EasyLearning Objective: 08-04 Explain how sample size affects the standard error.Topic: Confidence Interval for a Mean () with Unknown

13.As n increases, the width of the confidence interval will decrease, ceteris paribus.TRUEThe standard error is the standard deviation divided by the square root of the sample size.


14.As n increases, the standard error decreases.TRUEThe standard error is the standard deviation divided by the square root of the sample size.

AACSB: AnalyticBlooms: RememberDifficulty: 2 MediumLearning Objective: 08-04 Explain how sample size affects the standard error.Topic: Sample Mean and the Central Limit Theorem

15.A higher confidence level leads to a narrower confidence interval, ceteris paribus.FALSEHigher confidence requires more uncertainty (a wider interval). For example, z.025 = 1.960 (for 95 percent confidence) gives a wider interval than z.05 = 1.645 (for 90 percent confidence). The proffered statement would also hold true for the Student's t distribution.


16.When the sample standard deviation is used to construct a confidence interval for the mean, we would use the Student's t distribution instead of the normal distribution.TRUEWe should use t when the population variance is unknown.


17.As long as the sample is more than one item, the standard error of the sample mean will be smaller than the standard deviation of the population.TRUEThe standard error is the standard deviation divided by the square root of the sample size.

AACSB: AnalyticBlooms: UnderstandDifficulty: 1 EasyLearning Objective: 08-04 Explain how sample size affects the standard error.Topic: Sample Mean and the Central Limit Theorem

18.For a sample size of 20, a 95 percent confidence interval using the t distribution would be wider than one constructed using the z distribution.TRUEStudent's t is always larger than z for the same level of confidence.

AACSB: AnalyticBlooms: RememberDifficulty: 2 MediumLearning Objective: 08-06 Know when to use Student's t instead of z to estimate .Topic: Confidence Interval for a Mean () with Unknown

19.In constructing a confidence interval for a mean, the width of the interval is dependent on the sample size, the confidence level, and the population standard deviation.TRUEThe confidence interval depends on all of these.

AACSB: AnalyticBlooms: RememberDifficulty: 2 MediumLearning Objective: 08-05 Construct a 90; 95; or 99 percent confidence interval for .Topic: Confidence Interval for a Mean () with Known

20.In constructing confidence intervals, it is conservative to use the z distribution when n 30.FALSEWhile t and z may be similar for large samples, it is more conservative to use t.


21.The Central Limit Theorem can be applied to the sample proportion.TRUEWe are sampling a Bernoulli population, but the CLT still applies.

AACSB: AnalyticBlooms: RememberDifficulty: 1 EasyLearning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .Topic: Confidence Interval for a Proportion ()

22.The distribution of the sample proportion p = x/n is normal when n 30.FALSEWe want at least 10 successes and 10 failures to assume that p is normally distributed.

AACSB: AnalyticBlooms: UnderstandDifficulty: 1 EasyLearning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .Topic: Confidence Interval for a Proportion ()

23.The standard deviation of the sample proportion p = x/n increases as n increases.FALSEThe proffered statement is backwards because n is in the denominator of [p(1 - p)/n]1/2.


24.A 95 percent confidence interval constructed around p will be wider than a 90 percent confidence interval.TRUEHigher confidence requires more uncertainty (a wider interval).

AACSB: AnalyticBlooms: UnderstandDifficulty: 1 EasyLearning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .Topic: Confidence Interval for a Proportion ()

25.The sample proportion is always the midpoint of a confidence interval for the population proportion.TRUEThe interval is p z[p(1 - p)/n]1/2.

AACSB: AnalyticBlooms: RememberDifficulty: 2 MediumLearning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .Topic: Confidence Interval for a Proportion ()

26.The standard error of the sample proportion is largest when = .50.TRUEThe value of [(1 - )/n]1/2 is smaller for any value less than = .50.

AACSB: AnalyticBlooms: UnderstandDifficulty: 2 MediumLearning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .Topic: Confidence Interval for a Proportion ()

27.The standard error of the sample proportion does not depend on the confidence level.TRUEThe standard error of p is [(1 - )/n]1/2.


28.To narrow the confidence interval for , we can either increase n or decrease the level of confidence.TRUEThe interval is p z[p(1 - p)/n]1/2.


29.Ceteris paribus, the narrowest confidence interval for is achieved when p = .50.FALSEThe value of [p(1 - p)/n]1/2 is smaller for any value less than = .50.


30.The statistic p = x/n may be assumed normally distributed when np 10 and n(1 - p) 10.TRUEWe want at least 10 successes and 10 failures in the sample to assume normality of p.


31.The Student's t distribution is always symmetric and bell-shaped, but its tails lie above the normal.TRUEStudent's t resembles a normal, but its PDF is above the normal PDF in the tails.

AACSB: AnalyticBlooms: RememberDifficulty: 2 MediumLearning Objective: 08-06 Know when to use Student's t instead of z to estimate .Topic: Confidence Interval for a Mean () with Unknown

32.The confidence interval half-width when = .50 is called the margin of error.TRUEPollsters use this definition.


33.Based on the Rule of Three, if no events occur in n independent trials we can set the upper 95 percent confidence bound at 3/n.TRUEWe need a special rule because when p = 0 we can't apply the usual formula p z[p(1 - p)/n]1/2.


34.The sample standard deviation s is halfway between the lower and upper confidence limits for the population (i.e., the confidence interval is symmetric around s).FALSEThe chi-square distribution is not symmetric.

AACSB: AnalyticBlooms: ApplyDifficulty: 3 HardLearning Objective: 08-10 Construct a confidence interval for a variance (optional).Topic: Confidence Interval for a Population Variance, 2 (Optional)

35.In a sample size calculation, if the confidence level decreases, the size of the sample needed will increase.FALSEReduced confidence allows a smaller sample.

AACSB: AnalyticBlooms: UnderstandDifficulty: 2 MediumLearning Objective: 08-09 Calculate sample size to estimate a mean or proportion.Topic: Sample Size Determination for a Mean

36.To calculate the sample size needed for a survey to estimate a proportion, the population standard deviation must be known.FALSEFor a proportion, the sample size formula requires not .

AACSB: AnalyticBlooms: ApplyDifficulty: 2 MediumLearning Objective: 08-09 Calculate sample size to estimate a mean or proportion.Topic: Sample Size Determination for a Proportion

37.Assuming that = .50 is a quick and conservative approach to use in a sample size calculation for a proportion.TRUEAssuming that = .50 is quick and safe (but may give a larger sample than is needed).

AACSB: AnalyticBlooms: RememberDifficulty: 2 MediumLearning Objective: 08-09 Calculate sample size to estimate a mean or proportion.Topic: Sample Size Determination for a Proportion

38.To estimate the required sample size for a proportion, one method is to take a small pilot sample to estimate and then apply the sample size formula.TRUEThis is a common method, but assuming that = .50 is quicker and safer.

AACSB: AnalyticBlooms: RememberDifficulty: 2 MediumLearning Objective: 08-09 Calculate sample size to estimate a mean or proportion.Topic: Sample Size Determination for a Proportion

39.To estimate , you typically need a sample size equal to at least 5 percent of your population.FALSEThe sample size n bears no necessary relation to N.


40.To estimate a proportion with a 4 percent margin of error and a 95 percent confidence level, the required sample size is over 800.FALSEn = (z/E)2()(1 - ) = (1.96/.04)2(.50)(1 - .50) = 600.25.


41.Approximately 95 percent of the population X values will lie within the 95 percent confidence interval for the mean.FALSEThe confidence interval is for the true mean, not for individual X values.

AACSB: AnalyticBlooms: ApplyDifficulty: 2 MediumLearning Objective: 08-05 Construct a 90; 95; or 99 percent confidence interval for .Topic: Confidence Interval for a Mean () with Known

42.A 99 percent confidence interval has more confidence but less precision than a 95 percent confidence interval.TRUEThe higher confidence level widens the interval so it is less precise.

AACSB: AnalyticBlooms: ApplyDifficulty: 2 MediumLearning Objective: 08-05 Construct a 90; 95; or 99 percent confidence interval for .Topic: Confidence Interval for a Mean () with Known

43.Sampling variation is not controllable by the statistician.TRUESampling variation is inevitable.

AACSB: AnalyticBlooms: RememberDifficulty: 1 EasyLearning Objective: 08-01 Define sampling error; parameter; and estimator.Topic: Sampling Variation

44.The sample mean is not a random variable when the population parameters are known.FALSEThe sample mean is a random variable regardless of what we know about the population.

AACSB: AnalyticBlooms: RememberDifficulty: 1 EasyLearning Objective: 08-01 Define sampling error; parameter; and estimator.Topic: Sampling Variation

45.The finite population correction factor (FPCF) can be ignored if n = 7 and N = 700.TRUEThe FPCF has a negligible effect when the sample is less than 5 percent of the population.

AACSB: AnalyticBlooms: ApplyDifficulty: 2 MediumLearning Objective: 08-08 Construct confidence intervals for finite populations.Topic: Estimating from Finite Populations

46.In constructing a confidence interval, the finite population correction factor (FPCF) can be ignored if samples of 12 items are drawn from a population of 300 items.TRUEThe FPCF has a negligible effect when the sample is less than 5 percent of the population.


47.The finite population correction factor (FPCF) can be ignored when the sample size is large relative to the population size.TRUEThe FPCF has a negligible effect when n is small relative to N.


Multiple Choice Questions48.A sampling distribution describes the distribution of:

A.a parameter.

B.a statistic.

C.either a parameter or a statistic.

D.neither a parameter nor a statistic.

A statistic has a sampling distribution.

AACSB: AnalyticBlooms: RememberDifficulty: 1 EasyLearning Objective: 08-03 State the Central Limit Theorem for a mean.Topic: Sample Mean and the Central Limit Theorem

49.As the sample size increases, the standard error of the mean:

A.increases.

B.decreases.

C.may increase or decrease.

The standard error of the mean is /(n)1/2.

AACSB: AnalyticBlooms: RememberDifficulty: 1 EasyLearning Objective: 08-03 State the Central Limit Theorem for a mean.Topic: Sample Mean and the Central Limit Theorem

50.Which statement is most nearly correct, other things being equal?

A.Doubling the sample size will cut the standard error of the mean in half.

B.The standard error of the mean depends on the population size.

C.Quadrupling the sample size roughly halves the standard error of the mean.

D.The standard error of the mean depends on the confidence level.

The standard error of the mean is /(n)1/2 so replacing n by 4n would cut the SEM in half.

AACSB: AnalyticBlooms: UnderstandDifficulty: 2 MediumLearning Objective: 08-04 Explain how sample size affects the standard error.Topic: Sample Mean and the Central Limit Theorem

51.The width of a confidence interval for is not affected by:

A.the sample size.

B.the confidence level.

C.the standard deviation.

D.the sample mean.

The mean is not used in calculating the width of the confidence interval z/(n)1/2.


52.The Central Limit Theorem (CLT) implies that:

A.the population will be approximately normal if n 30.

B.repeated samples must be taken to obtain normality.

C.the distribution of the mean is approximately normal for large n.

D.the mean follows the same distribution as the population.

The sampling distribution of the mean is asymptotically normal for any population.

AACSB: AnalyticBlooms: UnderstandDifficulty: 2 MediumLearning Objective: 08-03 State the Central Limit Theorem for a mean.Topic: Sample Mean and the Central Limit Theorem

53.The owner of Limp Pines Resort wanted to know the average age of its clients. A random sample of 25 tourists is taken. It shows a mean age of 46 years with a standard deviation of 5 years. The width of a 98 percent CI for the true mean client age is approximately:

A. 1.711 years.

B. 2.326 years.

C. 2.492 years.

D. 2.797 years.

The width is ts/(n)1/2 = (2.492)(5)/(25)1/2 = 2.492.

AACSB: AnalyticBlooms: ApplyDifficulty: 2 MediumLearning Objective: 08-06 Know when to use Student's t instead of z to estimate .Topic: Confidence Interval for a Mean () with Unknown

54.In constructing a confidence interval for a mean with unknown variance with a sample of 25 items, Bob used z instead of t. "Well, at least my interval will be wider than necessary, so it was a conservative error," said he. Is Bob's statement correct?

A.Yes.

B.No.

C.It depends on .

z is always smaller than t (ceteris paribus) so the interval would be narrower than is justified.


55.A random sample of 16 ATM transactions at the Last National Bank of Flat Rock revealed a mean transaction time of 2.8 minutes with a standard deviation of 1.2 minutes. The width (in minutes) of the 95 percent confidence interval for the true mean transaction time is:

A. 0.639

B. 0.588

C. 0.300

D. 2.131

The width is ts/(n)1/2 = (2.131)(1.2)/(16)1/2 = 0.639.


56.We could narrow a 95 percent confidence interval by:

A.using 99 percent confidence.

B.using a larger sample.

C.raising the standard error.

A larger sample would narrow the interval width z/(n)1/2.


57.The owner of Torpid Oaks B&B wanted to know the average distance its guests had traveled. A random sample of 16 guests showed a mean distance of 85 miles with a standard deviation of 32 miles. The 90 percent confidence interval (in miles) for the mean is approximately:

A.(71.0, 99.0)

B.(71.8, 98.2)

C.(74.3, 95.7)

D.(68.7, 103.2)

The interval is 85 ts/(n)1/2 or 85 (1.753)(32)/(16)1/2 with d.f = 15 (don't use z).


58.A highway inspector needs an estimate of the mean weight of trucks crossing a bridge on the interstate highway system. She selects a random sample of 49 trucks and finds a mean of 15.8 tons with a sample standard deviation of 3.85 tons. The 90 percent confidence interval for the population mean is:

A.14.72 to 16.88 tons.

B.14.90 to 16.70 tons.

C.14.69 to 16.91 tons.

D.14.88 to 16.72 tons.

The interval is 15.8 ts/(n)1/2 or 15.8 (1.677)(3.85)/(49)1/2 using d.f. = 48 (don't use z).


59.To determine a 72 percent level of confidence for a proportion, the value of z is approximately:

A. 1.65

B. 0.77

C. 1.08

D. 1.55

Look up the z value that puts 14 percent in each tail.

AACSB: AnalyticBlooms: ApplyDifficulty: 2 MediumLearning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .Topic: Confidence Interval for a Proportion ()

60.To estimate the average annual expenses of students on books and class materials a sample of size 36 is taken. The sample mean is $850 and the sample standard deviation is $54. A 99 percent confidence interval for the population mean is:

A.$823.72 to $876.28

B.$832.36 to $867.64

C.$826.82 to $873.18

D.$825.48 to $874.52

The interval is 850 ts/(n)1/2 or 850 (2.724)(54)/(36)1/2 with d.f = 35 (don't use z).

AACSB: AnalyticBlooms: ApplyDifficulty: 2 MediumLearning Objective: 08-05 Construct a 90; 95; or 99 percent confidence interval for .Topic: Confidence Interval for a Mean () with Unknown

61.In constructing a 95 percent confidence interval, if you increase n to 4n, the width of your confidence interval will (assuming other things remain the same) be:

A.about 25 percent of its former width.

B.about two times wider.

C.about 50 percent of its former width.

D.about four times wider.

The standard error of the mean is /(n)1/2 so replacing n by 4n would cut the SEM in half.

AACSB: AnalyticBlooms: ApplyDifficulty: 2 MediumLearning Objective: 08-04 Explain how sample size affects the standard error.Topic: Confidence Interval for a Mean () with Known

62.Which of the following is not a characteristic of the t distribution?

A.It is a continuous distribution.

B.It has a mean of 0.

C.It is a symmetric distribution.

D.It approaches z as degrees of freedom decrease.

It approaches z as degrees of freedom increase.

AACSB: AnalyticBlooms: ApplyDifficulty: 1 EasyLearning Objective: 08-06 Know when to use Student's t instead of z to estimate .Topic: Confidence Interval for a Mean () with Unknown

63.Which statement is incorrect? Explain.

A.If p = .50 and n = 100, the standard error of the sample proportion is .05.

B.In a sample size calculation for estimating , it is conservative to assume = .50.

C.If n = 250 and p = .06, we cannot assume normality in a confidence interval for .

Normality of p may be assumed because np = 15 and n(1 - p) = 235.


64.What is the approximate width of a 90 percent confidence interval for the true population proportion if there are 12 successes in a sample of 25?

A. .196

B. .164

C. .480

D. .206

The interval width is z[p(1 - p)/n]1/2 = (1.645)[(.48)(.52)/25]1/2.


65.A poll showed that 48 out of 120 randomly chosen graduates of California medical schools last year intended to specialize in family practice. What is the width of a 90 percent confidence interval for the proportion that plan to specialize in family practice?

A. .0447

B. .0736

C. .0876

D. .0894



66.What is the approximate width of an 80 percent confidence interval for the true population proportion if there are 12 successes in a sample of 80?

A. .078

B. .066

C. .051

D. .094



67.A random sample of 160 commercial customers of PayMor Lumber revealed that 32 had paid their accounts within a month of billing. The 95 percent confidence interval for the true proportion of customers who pay within a month would be:

A.0.148 to 0.252

B.0.138 to 0.262

C.0.144 to 0.256

D.0.153 to 0.247

The interval is p z[p(1 - p)/n]1/2 = .20 (1.960)[(.20)(.80)/160]1/2.


68.A random sample of 160 commercial customers of PayMor Lumber revealed that 32 had paid their accounts within a month of billing. Can normality be assumed for the sample proportion?

A.Yes.

B.No.

C.Need more information to say.

Yes, because there were at least 10 "successes" and at least 10 "failures" in the sample.

AACSB: AnalyticBlooms: ApplyDifficulty: 1 EasyLearning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .Topic: Confidence Interval for a Proportion ()

69.The conservative sample size required for a 95 percent confidence interval for with an error of 0.04 is:

A.271.

B.423.

C.385.

D.601.

n = (z/E)2()(1 - ) = (1.96/.04)2(.50)(1 - .50) = 600.25 (round up).


70.Last week, 108 cars received parking violations in the main university parking lot. Of these, 27 had unpaid parking tickets from a previous violation. Assuming that last week was a random sample of all parking violators, find the 95 percent confidence interval for the percentage of parking violators that have prior unpaid parking tickets.

A.18.1 to 31.9 percent.

B.16.8 to 33.2 percent.

C.15.3 to 34.7 percent.

D.19.5 to 30.5 percent.



71.In a random sample of 810 women employees, it is found that 81 would prefer working for a female boss. The width of the 95 percent confidence interval for the proportion of women who prefer a female boss is:

A. .0288

B. .0105

C. .0207

D. .0196

The width is z[p(1 - p)/n]1/2 or (1.960)[(.10)(.90)/810]1/2 or .0207.


72.Jolly Blue Giant Health Insurance (JBGHI) is concerned about rising lab test costs and would like to know what proportion of the positive lab tests for prostate cancer are actually proven correct through subsequent biopsy. JBGHI demands a sample large enough to ensure an error of 2 percent with 90 percent confidence. What is the necessary sample size?

A.4,148

B.2,401

C.1,692

D.1,604

n = (z/E)2()(1 - ) = (1.645/.02)2(.50)(1 - .50) = 1691.3 (round up).


73.A university wants to estimate the average distance that commuter students travel to get to class with an error of 3 miles and 90 percent confidence. What sample size would be needed, assuming that travel distances are normally distributed with a range of X = 0 to X = 50 miles, using the Empirical Rule 3 to estimate .

A.About 28 students

B.About 47 students

C.About 30 students

D.About 21 students

Using = (50 - 0)/6 = 8.333, we get n = [z/E]2 = [(1.645)(8.333)/3]2 = 20.9 (round up).

AACSB: AnalyticBlooms: ApplyDifficulty: 3 HardLearning Objective: 08-09 Calculate sample size to estimate a mean or proportion.Topic: Sample Size Determination for a Mean

74.A financial institution wishes to estimate the mean balances owed by its credit card customers. The population standard deviation is $300. If a 99 percent confidence interval is used and an interval of $75 is desired, how many cardholders should be sampled?

A.3382

B.629

C.87

D.107

n = [z/E]2 = [(2.576)(300)/75]2 = 106.2 (round up).

AACSB: AnalyticBlooms: ApplyDifficulty: 2 MediumLearning Objective: 08-09 Calculate sample size to estimate a mean or proportion.Topic: Sample Size Determination for a Mean

75.A company wants to estimate the time its trucks take to drive from city A to city B. The standard deviation is known to be 12 minutes. What sample size is required in order that error will not exceed 2 minutes, with 95 percent confidence?

A.12 observations

B.139 observations

C.36 observations

D.129 observations

n = [z/E]2 = [(1.960)(12)/2]2 = 138.3 (round up).


76.In a large lecture class, the professor announced that the scores on a recent exam were normally distributed with a range from 51 to 87. Using the Empirical Rule 3 to estimate , how many students would you need to sample to estimate the true mean score for the class with 90 percent confidence and an error of 2?

A.About 17 students

B.About 35 students

C.About 188 students

D.About 25 students

Using = (87 - 51)/6 = 6, we get n = [z/E]2 = [(1.645)(6)/2]2 = 24.35 (round up).


77.Using the conventional polling definition, find the margin of error for a customer satisfaction survey of 225 customers who have recently dined at Applebee's.

A. 5.0 percent

B. 4.2 percent

C. 7.1 percent

D. 6.5 percent

The margin of error is z[(1 - )/n]1/2 or (1.960)[(.50)(.50)/225]1/2 or .065.


78.A marketing firm is asked to estimate the percent of existing customers who would purchase a "digital upgrade" to their basic cable TV service. The firm wants 99 percent confidence and an error of 5 percent. What is the required sample size (to the next higher integer)?

A.664

B.625

C.801

D.957

n = (z/E)2()(1 - ) = (2.576/.05)2(.50)(1 - .50) = 663.6 (round up).


79.An airport traffic analyst wants to estimate the proportion of daily takeoffs by small business jets (as opposed to commercial passenger jets or other aircraft) with an error of 4 percent with 90 percent confidence. What sample size should the analyst use?

A.385

B.601

C.410

D.423

n = (z/E)2()(1 - ) = (1.645/.04)2(.50)(1 - .50) = 422.8 (round up).


80.Ersatz Beneficial Insurance wants to estimate the cost of damage to cars due to accidents. The standard deviation of the cost is known to be $200. They want to estimate the mean cost using a 95 percent confidence interval within $10. What is the minimum sample size n?

A.1083

B.4002

C.1537

D.2301

n = [z/E]2 = [(1.960)(200)/10]2 = 1536.6 (round up).


81.Professor York randomly surveyed 240 students at Oxnard University and found that 150 of the students surveyed watch more than 10 hours of television weekly. Develop a 95 percent confidence interval to estimate the true proportion of students who watch more than 10 hours of television each week. The confidence interval is:

A..533 to .717

B..564 to .686

C..552 to .698

D..551 to .739



82.Professor York randomly surveyed 240 students at Oxnard University and found that 150 of the students surveyed watch more than 10 hours of television weekly. How many additional students would Professor York have to sample to estimate the proportion of all Oxnard University students who watch more than 10 hours of television each week within 3 percent with 99 percent confidence?

A.761

B.1001

C.1489

D.1728

Using p = .625 we get n = (z/E)2()(1 - ) = (2.576/.03)2(.625)(.375) = 1728.06 (round up).

AACSB: AnalyticBlooms: ApplyDifficulty: 3 HardLearning Objective: 08-09 Calculate sample size to estimate a mean or proportion.Topic: Sample Size Determination for a Proportion

83.The sample proportion is in the middle of the confidence interval for the population proportion:

A.in any sample.

B.only if the samples are large.

C.only if is not too far from .50.

The interval is p z[p(1 - p)/n]1/2.



A.6.262, 27.49

B.6.908, 28.85

C.5.629, 26.12

D.7.261, 25.00

Using d.f. = n - 1 = 15, we get 2L = 6.262 and 2U = 27.49 from Appendix E.

AACSB: AnalyticBlooms: ApplyDifficulty: 2 MediumLearning Objective: 08-10 Construct a confidence interval for a variance (optional).Topic: Confidence Interval for a Population Variance, 2 (Optional)


A.6.262, 27.49

B.6.908, 28.85

C.3.940, 18.31

D.3.247, 20.48

d.f. = n - 1 = 10, we get 2L = 3.940 and 2U = 18.31 from Appendix E.



A.6.262, 27.49

B.5.697, 35.72

C.5.629, 26.12

D.7.261, 25.00

d.f. = n - 1 = 17, we get 2L = 5.697 and 2U = 35.72 from Appendix E.


87.Which of the following statements is most nearly correct, other things being equal?

A.Using Student's t instead of z makes a confidence interval narrower.

B.The table values of z and t are about the same when the mean is large.

C.For a given confidence level, the z value is always smaller then the t value.

D.Student's t is rarely used because it is more conservative to use z.

As n increases, t approaches z, but t is always larger than z.

AACSB: AnalyticBlooms: UnderstandDifficulty: 2 MediumLearning Objective: 08-04 Explain how sample size affects the standard error.Topic: Confidence Interval for a Mean () with Unknown

88.The Central Limit Theorem (CLT):

A.applies only to samples from normal populations.

B.applies to any population.

C.applies best to populations that are skewed.

D.applies only when and are known.

The appeal of the CLT is that is applies to populations of any shape.



A.12 successes in a sample of 72 items

B.8 successes in a sample of 40 items

C.6 successes in a sample of 200 items

D.4 successes in a sample of 500 items

We prefer at least 10 "successes" and at least 10 "failures" to assume that p is normal.

AACSB: AnalyticBlooms: RememberDifficulty: 2 MediumLearning Objective: 08-03 State the Central Limit Theorem for a mean.Topic: Confidence Interval for a Proportion ()


A.n = 100, = .06

B.n = 250, = .02

C.n = 30, = .50

D.n = 500, = .01

We want n 10 and n(1 - ) 10 to assume that p is normal.

AACSB: AnalyticBlooms: RememberDifficulty: 2 MediumLearning Objective: 08-03 State the Central Limit Theorem for a mean.Topic: Confidence Interval for a Proportion ()


A.75

B.35

C.58

D.113

n = [z/E]2 = [(1.960)(12)/4]2 = 34.6 (round up).

AACSB: AnalyticBlooms: RememberDifficulty: 2 MediumLearning Objective: 08-03 State the Central Limit Theorem for a mean.Topic: Sample Size Determination for a Mean


A.426

B.512

C.267

D.188

n = [z/E]2 = [(1.645)(25)/3]2 = 187.9 (round up).

AACSB: AnalyticBlooms: RememberDifficulty: 2 MediumLearning Objective: 08-03 State the Central Limit Theorem for a mean.Topic: Sample Size Determination for a Mean

93.Sampling error can be avoided:

A.by using an unbiased estimator.

B.by eliminating nonresponses (e.g., older people).

C.by no method under the statistician's control.

D.either by using an unbiased estimator or by eliminating nonresponse.

Sampling error occurs in any random sample used to estimate an unknown parameter.


94.A consistent estimator for the mean:

A.converges on the true parameter as the variance increases.

B.converges on the true parameter as the sample size increases.

C.consistently follows a normal distribution.

D.is impossible to obtain using real sample data.

The variance becomes smaller and the estimator approaches the parameter as n increases.

AACSB: AnalyticBlooms: RememberDifficulty: 2 MediumLearning Objective: 08-02 Explain the desirable properties of estimators.Topic: Estimators and Sampling Error

95.Concerning confidence intervals, which statement is most nearly correct?

A.We should use z instead of t when n is large.

B.We use the Student's t distribution when is unknown.

C.We use the Student's t distribution to narrow the confidence interval.

Student's t distribution widens the confidence interval when is unknown.

AACSB: AnalyticBlooms: UnderstandDifficulty: 2 MediumLearning Objective: 08-06 Know when to use Student's t instead of z to estimate .Topic: Confidence Interval for a Mean () with Unknown

96.The standard error of the mean decreases when:

A.the sample size decreases.

B.the standard deviation increases.

C.the standard deviation decreases or n increases.

D.the population size decreases.

The standard error of the mean /(n1/2) depends on n and .

AACSB: AnalyticBlooms: UnderstandDifficulty: 2 MediumLearning Objective: 08-04 Explain how sample size affects the standard error.Topic: Confidence Interval for a Mean () with Known

97.For a given sample size, the higher the confidence level, the:

A.more accurate the point estimate.

B.smaller the standard error.

C.smaller the interval width.

D.greater the interval width.

To have more confidence, we must widen the interval. For example, z.025 = 1.960 (for 95 percent confidence) gives a wider interval than z.05 = 1.645 (for 90 percent confidence). The proffered statement would also be true for the Student's t distribution.


98.A sample is taken and a confidence interval is constructed for the mean of the distribution. At the center of the interval is always which value?

A.The sample mean

B.The population mean

C.Neither nor since with a sample anything can happen

D.Both and as long as there are not too many outliers

The confidence interval for the mean is symmetric around the sample mean.


99.If a normal population has parameters = 40 and = 8, then for a sample size n = 4:

A.the standard error of the sample mean is approximately 2.

B.the standard error of the sample mean is approximately 4.

C.the standard error of the sample mean is approximately 8.

D.the standard error of the sample mean is approximately 10.

The standard error is /(n1/2) = (8)/(41/2) = 4.

AACSB: AnalyticBlooms: ApplyDifficulty: 1 EasyLearning Objective: 08-04 Explain how sample size affects the standard error.Topic: Confidence Interval for a Mean () with Known

Short Answer Questions100.On the basis of a survey of 545 television viewers, a statistician has constructed a confidence interval and estimated that the proportion of people who watched the season premiere of Glee is between .16 and .24. What level of confidence did the statistician use in constructing this interval? Explain carefully, showing all steps in your reasoning.

We solve to get z = 2.33, which corresponds approximately to a 98 percent confidence level.

Feedback: The confidence interval is

and the interval half-width is .04 so we set

and p = .20 (the midpoint of the interval) to solve for

= 2.33 which corresponds approximately to a 98 percent confidence level.

AACSB: AnalyticBlooms: EvaluateDifficulty: 3 HardLearning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .Topic: Confidence Interval for a Proportion ()

101.Read the news story below. Using the 95 percent confidence level, what sample size would be needed to estimate the true proportion of stores selling cigarettes to minors with an error of 3 percent? Explain carefully, showing all steps in your reasoning.

=

= 813.5, or 814 (rounded up), using the sample proportion because it is available (instead of assuming that = .50).

Feedback:

=

= 813.5, or 814 (rounded up). We use the sample proportion because it is available, instead of assuming that = .50.


102.In a survey, 858 out of 2600 homeowners said they expected good economic conditions to continue for the next 12 months. Construct a 95 percent confidence interval for "good times" in the next 12 months.

The confidence interval is .3119 < < .3481.

Feedback:

or

or

or .33 .0181, so the confidence interval is .3119 < < .3481.


103.Fulsome University has 16,059 students. In a sample of 200 students, 12 were born outside the United States. Construct a 95 percent confidence interval for the true population proportion. How large a sample is needed to estimate the true proportion of Fulsome students who were born outside the United States with an error of 2.5 percent and 95 percent confidence? Show your work and explain fully.

We have sampled less than 5 percent of the population, so the FPCF is unnecessary (i.e., we can ignore the population size. The 95 percent confidence interval is p z.025[p(1 - p)/n]1/2 = .06 (1.960)[(.06)(.94)/200]1/2 or .06 .032914 or .027 < < .093. To reduce the error to .025, the required sample size is

or

= 346.7, or n = 347 (rounded up). We can use the sample value for p so we do not need to assume that = .50.

Feedback: The 95 percent confidence interval is p z.025[p(1 - p)/n]1/2 = .06 (1.960)[(.06)(.94)/200]1/2 or .06 .032914 or .027 < < .093. To reduce the error to .025, the required sample size is

or

= 346.7, or n = 347 (rounded up). We have a sample value for p so we do not need to assume that = .50. If you did assume = .50, you would get an unnecessarily large required sample since the preliminary sample indicates that is not .50. The sample does not exceed 5 percent of the population size, so the finite population correction would make little difference.

AACSB: AnalyticBlooms: ApplyDifficulty: 3 HardLearning Objective: 08-09 Calculate sample size to estimate a mean or proportion.Topic: Sample Size Determination for a Proportion

104.List differences and similarities between Student's t and the standard normal distribution.

Both are bell-shaped and symmetric, but the Student's t distribution lies below the standard normal in the middle, and its tails are above the standard normal.

Feedback: They are both bell-shaped and symmetric. However, the Student's t distribution lies below the standard normal in the middle, and its tails are above the standard normal ("thicker" or "heavier" tails). Therefore, the value of Student's t for a given tail area will always be greater than the corresponding z value. We use the Student's t whenever the standard deviation is estimated from a sample, which is to say, most of the time.

AACSB: AnalyticBlooms: ApplyDifficulty: 1 EasyLearning Objective: 08-06 Know when to use Student's t instead of z to estimate .Topic: Confidence Interval for a Mean () with Unknown

105.Why does pose a problem for sample size calculation for a mean? How can be approximated when it is unknown?

Truehe formula for the sample size to estimate requires knowing . But because is unknown (we are trying to estimate it), then probably is unknown as well. There are several ways to estimate : (1) take a small preliminary sample and calculate the sample standard deviation s as an estimate of ; or (2) if the range is known, we can estimate = Range/6 because from the Empirical Rule 3 contains almost all of the data in a normal distribution (a sometimes doubtful assumption if there are outliers or a skewed population); or (3) we might have some value for from prior experience (e.g., a previous sample or historical data).

Feedback: The formula for the sample size to estimate requires knowing . But because is unknown (we are trying to estimate it), then probably is unknown as well. There are several ways to estimate : (1) take a small preliminary sample and calculate the sample standard deviation s as an estimate of ; or (2) if the range is known, we can estimate = Range/6 because from the Empirical Rule 3 contains almost all of the data in a normal distribution (a sometimes doubtful assumption if there are outliers or a skewed population); or (3) we might have some value for from prior experience (e.g., a previous sample or historical data).
