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
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(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?
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TI-83 Plus
TI-83 Plus
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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
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?
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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
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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?
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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
a
. 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?
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