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6-2 Basic Skills and Concepts

Finding Critical Values. In Exercises 1–4, find the critical value that corresponds to

the given confidence level.

1. 99% 2. 90%

3. 98% 4. 92%

5. Express the confidence interval 0.220 , p , 0.280 in the form of  E.

6. Express the confidence interval 0.456 , p , 0.496 in the form of  E.

7. Express the confidence interval (0.604, 0.704) in the form of  E.

8. Express the confidence interval 0.742  0.030 in the form of 2 E , p ,1E.

Interpreting Confidence Interval Limits. In Exercises 9–12, use the given confidence in-

terval limits to find the point estimate and the margin of error E.

9. (0.444, 0.484) 10. 0.278 , p , 0.338

11. 0.632 , p , 0.678 12. 0.887 , p , 0.927

Finding Margin of Error. In Exercises 13–16, assume that a sample is used to estimate

a population proportion p. Find the margin of error E that corresponds to the given

statistics and confidence level.

13. n 5 800, x 5 200, 95% confidence

14. n 5 1200, x 5 400, 99% confidence

15. 99% confidence; the sample size is 1000, of which 45% are successes.

16. 95% confidence; the sample size is 500, of which 80% are successes.

Constructing Confidence Intervals. In Exercises 17–20, use the sample data and confi-

dence level to construct the confidence interval estimate of the population proportion p.

17. n 5 400, x 5 300, 95% confidence

18. n 5 1200, x 5 200, 99% confidence

19. n 5 1655, x 5 176, 98% confidence

20. n 5 2001, x 5 1776, 90% confidence

Determining Sample Size. In Exercises 21–24, use the given data to find the minimum

sample size required to estimate a population proportion or percentage.

21. Margin of error: 0.060; confidence level: 99%; and unknown

22. Margin of error: 0.038; confidence level: 95%; and unknown

23. Margin of error: five percentage points; confidence level: 95%; from a prior study,

is estimated by the decimal equivalent of 18.5%.

24. Margin of error: three percentage points; confidence level: 90%; from a prior study,

is estimated by the decimal equivalent of 8%.

25. Interpreting Calculator Display The Insurance Institute of America wants to estimate

the percentage of drivers aged 18–20 who drive a car while impaired because of alco-

hol consumption. In a large study, 42,772 males aged 18–20 were surveyed, and 5.1%

of them said that they drove in the last month while being impaired from alcohol

a>2

280 CHAPTER 6 Estimates and Sample Sizes

4025_CH06_p264-331 01/02/04 9:13 AM Page 280

(based on data from “Prevalence of Alcohol-Impaired Driving,” by Liu, Siegel, et al.,

Journal of the American Medical Association, Vol. 277, No. 2). Using the sample data

and a 95% confidence level, the TI-83 Plus calculator display is as shown.

a. Write a statement that interprets the confidence interval.

b. Based on the preceding result, does alcohol-impaired driving appear to be a prob-

lem for males aged 18–20? (All states now prohibit the sale of alcohol to persons

under the age of 21.)

c. When setting insurance rates for male drivers aged 18–24, what percentage of

alcohol-impaired driving would you use if you are working for the insurance com-

pany and you want to be conservative by using the likely worst case scenario?

26. Interpreting Calculator Display In 1920 only 35% of U.S. households had tele-

phones, but that rate is now much higher. A recent survey of 4276 randomly selected

households showed that 4019 of them had telephones (based on data from the U.S.

Census Bureau). Using those survey results and a 99% confidence level, the TI-83

Plus calculator display is as shown.

a. Write a statement that interprets the confidence interval.

b. Based on the preceding result, should pollsters be concerned about results from

surveys conducted by telephone?

27. Internet Shopping In a Gallup poll, 1025 randomly selected adults were surveyed and

29% of them said that they used the Internet for shopping at least a few times a year.

a. Find the point estimate of the percentage of adults who use the Internet for

shopping.

b. Find a 99% confidence interval estimate of the percentage of adults who use the

Internet for shopping.

c. If a traditional retail store wants to estimate the percentage of adult Internet shop-

pers in order to determine the maximum impact of Internet shoppers on its sales,

what percentage of Internet shoppers should be used?

28. Death Penalty Survey In a Gallup poll, 491 randomly selected adults were asked

whether they are in favor of the death penalty for a person convicted of murder, and

65% of them said that they were in favor.

a. Find the point estimate of the percentage of adults who are in favor of this death

penalty.

b. Find a 95% confidence interval estimate of the percentage of adults who are in fa-

vor of this death penalty.

c. Can we safely conclude that the majority of adults are in favor of this death

penalty? Explain.

29. Mendelian Genetics When Mendel conducted his famous genetics experiments with

peas, one sample of offspring consisted of 428 green peas and 152 yellow peas.

a. Find a 95% confidence interval estimate of the percentage of yellow peas.

b. Based on his theory of genetics, Mendel expected that 25% of the offspring peas

would be yellow. Given that the percentage of offspring yellow peas is not 25%,

do the results contradict Mendel’s theory? Why or why not?

30. Misleading Survey Responses In a survey of 1002 people, 701 said that they voted in

a recent presidential election (based on data from ICR Research Group). Voting

records show that 61% of eligible voters actually did vote.

a. Find a 99% confidence interval estimate of the proportion of people who say that

they voted.

b. Are the survey results consistent with the actual voter turnout of 61%? Why or

why not?

6-2 Estimating a Population Proportion 281

TI-83 Plus

4025_CH06_p264-331 01/02/04 9:13 AM Page 281

31. Drug Testing The drug Ziac is used to treat hypertension. In a clinical test, 3.2% of

221 Ziac users experienced dizziness (based on data from Lederle Laboratories).

a. Construct a 99% confidence interval estimate of the percentage of all Ziac users

who experience dizziness.

b. In the same clinical test, people in the placebo group didn’t take Ziac, but 1.8% of

them reported dizziness. Based on the result from part (a), what can we conclude

about dizziness as an adverse reaction to Ziac?

32. Smoking and College Education The tobacco industry closely monitors all surveys

that involve smoking. One survey showed that among 785 randomly selected subjects

who completed four years of college, 18.3% smoke (based on data from the American

Medical Association).

a. Construct the 98% confidence interval for the true percentage of smokers among

all people who completed four years of college.

b. Based on the result from part (a), does the smoking rate for those with four years

of college appear to be substantially different than the 27% rate for the general

population?

33. Sample Size for Internet Purchases Many states are carefully considering steps that

would help them collect sales taxes on items purchased through the Internet. How

many randomly selected sales transactions must be surveyed to determine the per-

centage that transpired over the Internet? Assume that we want to be 99% confident

that the sample percentage is within two percentage points of the true population per-

centage for all sales transactions.

34. Sample Size for Left-Handed Golfers As a manufacturer of golf equipment, the

Spalding Corporation wants to estimate the proportion of golfers who are left-handed.

(The company can use this information in planning for the number of right-handed

and left-handed sets of golf clubs to make.) How many golfers must be surveyed if we

want 99% confidence that the sample proportion has a margin of error of 0.025?

a. Assume that there is no available information that could be used as an estimate

of .

b. Assume that we have an estimate of found from a previous study that suggests

that 15% of golfers are left-handed (based on a USA Today report).

c. Assume that instead of using randomly selected golfers, the sample data are ob-

tained by asking TV viewers of the golfing channel to call an “800” phone number

to report whether they are left-handed or right-handed. How are the results

affected?

35. Sample Size for Motor Vehicle Ownership You have been hired by the Ford Motor

Company to do market research, and you must estimate the percentage of households

in which a vehicle is owned. How many households must you survey if you want to

be 94% confident that your sample percentage has a margin of error of three percent-

age points?

a. Assume that a previous study suggested that vehicles are owned in 86% of house-

holds.

b. Assume that there is no available information that can be used to estimate the per-

centage of households in which a vehicle is owned.

c. Assume that instead of using randomly selected households, the sample data are

obtained by asking readers of the Washington Post newspaper to mail in a survey

form. How are the results affected?

282 CHAPTER 6 Estimates and Sample Sizes

4025_CH06_p264-331 01/02/04 9:13 AM Page 282

36. Sample Size for Weapons on Campus Concerned about campus safety, college offi-

cials want to estimate the percentage of students who carry a gun, knife, or other such

weapon. How many randomly selected students must be surveyed in order to be 95%

confident that the sample percentage has a margin of error of three percentage points?

a. Assume that another study indicated that 7% of college students carry weapons

(based on a study by Cornell University).

b. Assume that there is no available information that can be used to estimate the per-

centage of college students carrying weapons.

37. Color Blindness In a study of perception, 80 men are tested and 7 are found to have

red green color blindness (based on data from USA Today).

a. Construct a 90% confidence interval estimate of the proportion of all men with this

type of color blindness.

b. What sample size would be needed to estimate the proportion of male red green

color blindness if we wanted 96% confidence that the sample proportion is in error

by no more than 0.03? Use the sample proportion as a known estimate.

c. Women have a 0.25% rate of red green color blindness. Based on the result from

part (a), can we safely conclude that women have a lower rate of red green color

blindness than men?

38. TV Ratings The CBS television show 60 Minutes has been successful for many

years. That show recently had a share of 20, meaning that among the TV sets in use,

20% were tuned to 60 Minutes (based on data from Nielsen Media Research). Assume

that this is based on a sample size of 4000 (typical for Nielsen surveys).

a. Construct a 97% confidence interval estimate of the proportion of all sets in use

that were tuned to 60 Minutes at the time of the broadcast.

b. What sample size would be required to estimate the percentage of sets tuned to 60

Minutes if we wanted 99% confidence that the sample percentage is in error by no

more than one-half of one percentage point? (Assume that we have no estimate of

the proportion.)

c. At the time of this particular 60 Minutes broadcast, ABC ran “Exposed: Pro

Wrestling,” and that show received a share of 11. Based on the result from part (a),

can we conclude that 60 Minutes had a greater proportion of viewers? Did profes-

sional wrestling really need to be exposed?

d. How is the confidence interval in part (a) affected if, instead of randomly selected

subjects, the survey data are based on 4000 television viewers volunteering to call

an “800” number to register their responses?

39. Cell Phones and Cancer A study of 420,000 Danish cell phone users found that 135

of them developed cancer of the brain or nervous system. Prior to this study of cell

phone use, the rate of such cancer was found to be 0.0340% for those not using cell

phones. The data are from the Journal of the National Cancer Institute.

a. Use the sample data to construct a 95% confidence interval estimate of the per-

centage of cell phone users who develop cancer of the brain or nervous system.

b. Do cell phone users appear to have a rate of cancer of the brain or nervous system

that is different from the rate of such cancer among those not using cell phones?

Why or why not?

40. Pilot Fatalities Researchers studied crashes of general aviation (noncommercial and

nonmilitary) airplanes and found that pilots died in 5.2% of 8411 crash landings

(based on data from “Risk Factors for Pilot Fatalities in General Aviation Airplane

6-2 Estimating a Population Proportion 283

4025_CH06_p264-331 01/02/04 9:13 AM Page 283

Crash Landings,” by Rostykus, Cummings, and Mueller, Journal of the American

Medical Association, Vol. 280, No. 11).

a. Construct a 95% confidence interval estimate of the percentage of pilot deaths in

all general aviation crashes.

b. Among crashes with an explosion or fire on the ground, the pilot fatality rate is

estimated by the 95% confidence interval of (15.5%, 26.9%). Is this result substan-

tially different from the result from part (a)? What can you conclude about an

explosion or fire as a risk factor?

c. In planning for the allocation of federal funds to help with medical examinations

of deceased pilots, what single percentage should be used? (We want to be reason-

ably sure that we have enough resources for the worst case scenario.)

41. Wearing Hunter Orange A study of hunting injuries and the wearing of “hunter”

orange clothing showed that among 123 hunters injured when mistaken for game, 6

were wearing orange. Among 1115 randomly selected hunters, 811 reported that they

routinely wear orange. The data are from the Centers for Disease Control.

a. Construct a 95% confidence interval estimate of the percentage of injured hunters

who are wearing orange.

b. Construct a 95% confidence interval estimate of the percentage of hunters who

routinely wear orange.

c. Do these results indicate that a hunter who wears orange is less likely to be injured

because of being mistaken for game? Why or why not?

42. Appearance Counts A Sales and Marketing Management survey included 651 sales

managers, and 94% of them said that being a sloppy dresser can make a sales repre-

sentative’s job more difficult. For that same group, 75% said that being an unstylish

dresser can make a sales representative’s job more difficult.

a. Construct a 90% confidence interval estimate of the percentage of sales managers

who say that being a sloppy dresser can make a sales representative’s job more

difficult.

b. Construct a 90% confidence interval estimate of the percentage of sales managers

who say that being an unstylish dresser can make a sales representative’s job more

difficult.

c. Given that sample proportions naturally vary, can we conclude that when sales

managers state reasons for a sales representative’s job becoming more difficult, the

percentage is higher for sloppy dressing than for unstylish dressing? Why or

why not?

43. Red M&M Candies Refer to Data Set 19 in Appendix B and find the sample propor-

tion of M&Ms that are red. Use that result to construct a 95% confidence interval

estimate of the population percentage of M&Ms that are red. Is the result consistent

with the 20% rate that is reported by the candy maker Mars?

44. Alcohol and Tobacco Use in Children’s Movies Refer to Data Set 7 in Appendix B.

a. Construct a 95% confidence interval estimate of the percentage of animated chil-

dren’s movies showing any tobacco use.

b. Construct a 95% confidence interval estimate of the percentage of animated chil-

dren’s movies showing any alcohol use.

c. Compare the preceding results. Does either tobacco or alcohol appear in a greater

percentage of animated children’s movies?

d. In using the results from parts (a) and (b) as measures of the depiction of unhealthy

habits, what important characteristic of the data is not included?

284 CHAPTER 6 Estimates and Sample Sizes

4025_CH06_p264-331 01/02/04 9:13 AM Page 284

6-2 Beyond the Basics

45. Probing for Precision An example of this section used the photo-cop survey data with

n 5 829 and 5 0.51 to construct the 95% confidence interval of 0.476 , p , 0.544.

However, cannot be exactly 0.51 because 51% of 829 people is 422.79 people,

which is not possible. The sample statistic of 51% has been rounded to the nearest

whole number. Find the minimum and maximum values of x for which x 829 is

rounded to 0.51, then construct the confidence intervals corresponding to those

two values of x. Do the results differ substantially from the confidence interval of

0.476 , p , 0.544 that was found using 0.51?

46. Using Finite Population Correction Factor This section presented Formulas 6-2 and

6-3, which are used for determining sample size. In both cases we assumed that the

population is infinite or very large and that we are sampling with replacement. When

we have a relatively small population with size N and sample without replacement,

we modify E to include the finite population correction factor shown here, and we

can solve for n to obtain the result given here. Use this result to repeat part (b) of Ex-

ercise 38, assuming that we limit our population to a town with 10,000 television sets

in use.

47. One-Sided Confidence Interval A one-sided confidence interval for p can be ex-

pressed as p ,1E or p .2E, where the margin of error E is modified by re-

placing with z

. If Air America wants to report an on-time performance of at least

x percent with 95% confidence, construct the appropriate one-sided confidence inter-

val and then find the percent in question. Assume that a simple random sample of 750

flights results in 630 that are on time.

48. Confidence Interval from Small Sample Special tables are available for finding

confidence intervals for proportions involving small numbers of cases, where the

normal distribution approximation cannot be used. For example, given x 5 3 suc-

cesses among n 5 8 trials, the 95% confidence interval found in Standard Probabil-

ity and Statistics Tables and Formulae (CRC Press) is 0.085 , p , 0.755. Find the

confidence interval that would result if you were to use the normal distribution in-

correctly as an approximation to the binomial distribution. Are the results reason-

ably close?

a>2

n 5

a>2

1 sN 2 1dE

E 5 z

a>2

N 2 n

N 2 1

6-3 Estimating a Population Mean: S Known 285

4025_CH06_p264-331 01/02/04 9:13 AM Page 285

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