SPEAKER WOOFER DRIVER MANUFACTURING
Step 2 The confidence interval for p is from
U. S. employees who play hooky
Solution We applied the one-proportionz-interval programs to the data, resulting in Output 12.2. Steps for generating that output are presented in Instructions 12.1.
Note to Excel users:For brevity, we have presented only the essential portions of the actual output.
MINITAB
OUTPUT 12.2 One-proportionz-interval on the data on playing hooky from work
TI-83/84 PLUS
EXCEL
As shown in Output 12.2, the required 95% confidence interval is from 0.175 to 0.225. We can be 95% confident that the percentage of all U.S. employees who play hooky is somewhere between 17.5% and 22.5%.
INSTRUCTIONS 12.1 Steps for generating Output 12.2 MINITAB
1 ChooseStat➤Basic Statistics➤1 Proportion. . . 2 Press the F3 key to reset the dialog box
3 SelectSummarized datafrom the drop-down list box 4 Click in theNumber of eventstext box and type202 5 Click in theNumber of trialstext box and type1010 6 Click theOptions. . . button
7 Click in theConfidence leveltext box and type95 8 SelectNormal approximationfrom theMethod
drop-down list box 9 ClickOKtwice
EXCEL
1 ChooseXLSTAT➤Parametric tests➤Tests for one proportion
2 Click the reset button in the lower left corner of the dialog box
3 Type202in theFrequencytext box 4 Type1010in theSample sizetext box 5 Click theOptionstab
6 Type5in theSignificance level (%)text box 7 ClickOK
TI-83/84 PLUS
1 PressSTAT, arrow over toTESTS, and pressALPHA➤A 2 Type202forxand pressENTER
3 Type1010fornand pressENTER
4 Type.95forC-Leveland pressENTERtwice
12.1 Confidence Intervals for One Population Proportion 575
Exercises 12.1
Understanding the Concepts and Skills
12.1 In a newspaper or magazine of your choice, find a statistical study that contains an estimated population proportion.
12.2 Why is statistical inference generally used to obtain informa- tion about a population proportion?
12.3 Is a population proportion a parameter or a statistic? What about a sample proportion? Explain your answers.
12.4 Regarding a population proportion:
a. What is it?
b. What symbol is used for it?
12.5 Regarding a sample proportion:
a. What is it?
b. What symbol is used for it?
12.6 Regarding the phrase “number of successes”:
a. For what is it an abbreviation?
b. What symbol is used for it?
12.7 For what is the phrase “number of failures” an abbreviation?
12.8 Explain the relationships among the sample proportion, the number of successes in the sample, and the sample size.
12.9 This exercise involves the use of an unrealistically small popu- lation to provide a concrete illustration for the exact distribution of a sample proportion. A population consists of three men and two women. The first names of the men are Jose, Pete, and Carlo; the first names of the women are Gail and Frances. Suppose that the specified attribute is “female.”
a. Determine the population proportion,p.
b. The first column of the following table provides the possible sam- ples of size 2, where each person is represented by the first let- ter of his or her first name; the second column gives the number of successes—the number of females obtained—for each sample;
and the third column shows the sample proportion. Complete the table.
Number of females Sample proportion
Sample x ˆp
J, G 1 0.5
J, P 0 0.0
J, C 0 0.0
J, F 1 0.5
G, P G, C G, F P, C P, F C, F
c. Construct a dotplot for the sampling distribution of the proportion for samples of size 2. Mark the position of the population propor- tion on the dotplot.
d. Use the third column of the table to obtain the mean of the vari- able ˆp.
e. Compare your answers from parts (a) and (d). Why are they the same?
12.10 Repeat parts (b)–(e) of Exercise 12.9 for samples of size 1.
12.11 Repeat parts (b)–(e) of Exercise 12.9 for samples of size 3.
(There are 10 possible samples.)
12.12 Repeat parts (b)–(e) of Exercise 12.9 for samples of size 4.
(There are five possible samples.)
12.13 Repeat parts (b)–(e) of Exercise 12.9 for samples of size 5.
12.14 Prerequisite to this exercise are Exercises 12.9–12.13. What do your graphs in parts (c) of those exercises illustrate about the impact of increasing sample size on sampling error? Explain your answer.
12.15 Man of the Tournament. From theWikipediapage “Cricket World Cup awards,” we found that since 1975, 40% of the ‘Man of the Final Match’ award has been won by the Australian Cricket Team.
a. Identify the population.
b. Identify the specified attribute.
c. Is the proportion 0.40 (40%) a population proportion or a sample proportion? Explain your answer.
12.16 Staying Single. From the U.S. Census Bureau document America’s Families and Living Arrangementsand an article inTime magazine, we found that, in 1963, 83.0% of American women be- tween the ages of 25 and 54 were married, compared to 64.6%
in 2010.
a. For 2010, identify the population.
b. For 2010, identify the specified attribute.
c. Under what circumstances is the proportion 0.646 a population proportion?
d. Under what circumstances is the proportion 0.646 a sample pro- portion?
12.17 Trustworthy. A certain poll asked Americans how many of the stated 19 companies are honest and trustworthy. Of 2,250 respon- dents, 49% said “None of the above”.
a. Determine the margin of error for a 95% confidence interval.
b. Without doing any calculations, indicate whether the margin of error is larger or smaller for a 90% confidence interval. Explain your answer.
12.18 Genetic Binge Eating. According to an article in Science News, binge eating has been associated with a mutation of the gene for a brain protein called melanocortin 4 receptor (MC4R). In one study, F. Horber of theHirslanden Clinicin Zurich and his colleagues genetically analyzed the blood of 469 obese people and found that 24 carried a mutated MC4R gene. Suppose that you want to esti- mate the proportion of all obese people who carry a mutated MC4R gene.
a. Determine the margin of error for a 90% confidence interval.
b. Without doing any calculations, indicate whether the margin of error is larger or smaller for a 95% confidence interval. Explain your answer.
In each of Exercises12.19–12.24, we have given a likely range for the observed value of a sample proportionp.ˆ
a. Based on the given range, identify the educated guess that should be used for the observed value ofp to calculate the required sam-ˆ ple size for a prescribed confidence level and margin of error.
b. Identify the observed values of the sample proportion that will yield a larger margin of error than the one specified if the edu- cated guess is used for the sample-size computation.
12.19 0.2 to 0.4 12.20 0.4 to 0.7
12.21 0.2 or less 12.22 0.7 or greater 12.23 0.4 or greater 12.24 0.7 or less
In each of Exercises12.25–12.30, we have given the number of suc- cesses and the sample size for a simple random sample from a popu- lation. In each case, do the following tasks.
a. Determine the sample proportion.
b. Decide whether using the one-proportion z-interval procedure is appropriate.
c. If appropriate, use the one-proportion z-interval procedure to find the confidence interval at the specified confidence level.
d. If appropriate, find the margin of error for the estimate of p and express the confidence interval in terms of the sample proportion and the margin of error.
12.25 x=8,n=40, 95% level 12.26 x=10,n=40, 90% level 12.27 x=35,n=50, 99% level 12.28 x=40,n=50, 95% level 12.29 x=52,n=100, 90% level 12.30 x=3,n=100, 99% level
In each of Exercises12.31–12.36, we have specified a margin of error and a confidence level. For each exercise, obtain a sample size that will ensure a margin of error of at most the one specified.
12.31 margin of error=0.01; confidence level=95%
12.32 margin of error=0.02; confidence level=95%
12.33 margin of error=0.02; confidence level=90%
12.34 margin of error=0.01; confidence level=90%
12.35 margin of error=0.03; confidence level=99%
12.36 margin of error=0.04; confidence level=99%
In Exercises12.37–12.42, we have specified the margin of errors and confidence levels from Exercises 12.31–12.36, respectively.
Additionally, we have, in each case, provided an educated guess for the observed value of the sample proportion. For each exercise, a. obtain a sample size that will ensure a margin of error of at most
the one specified (provided of course that the observed value of the sample proportion is further from 0.5 than the educated guess).
b. compare your answer to the corresponding one from Exer- cises 12.31–12.36 and explain the reason for the difference, if any.
12.37 margin of error =0.01; confidence level =95%; educated guess=0.3
12.38 margin of error =0.02; confidence level =95%; educated guess=0.6
12.39 margin of error=0.02; confidence level =90%; educated guess=0.1
12.40 margin of error=0.01; confidence level =90%; educated guess=0.9
12.41 margin of error=0.03; confidence level =99%; educated guess=0.5
12.42 margin of error=0.04; confidence level =99%; educated guess=0.5
In each of Exercises12.43–12.48, we have specified a margin of error, a confidence level, and a likely range for the observed value of
the sample proportion. For each exercise, obtain a sample size that will ensure a margin of error of at most the one specified (provided of course that the observed value of the sample proportion is further from 0.5 than the educated guess).
12.43 margin of error = 0.01; confidence level = 95%; likely range=0.2 to 0.4
12.44 margin of error = 0.02; confidence level = 95%; likely range=0.4 to 0.7
12.45 margin of error = 0.02; confidence level = 90%; likely range=0.2 or less
12.46 margin of error = 0.01; confidence level = 90%; likely range=0.7 or greater
12.47 margin of error = 0.03; confidence level = 99%; likely range=0.4 or greater
12.48 margin of error = 0.05; confidence level = 99%; likely range=0.6 or less
Applying the Concepts and Skills
In Exercises12.49–12.54, use Procedure 12.1 on page 570 to find the required confidence interval. Be sure to check the conditions for using that procedure.
12.49 In a trial of 280 patients who received 10 mg doses of a drug daily, 42 reported a headache as a side effect. Use the information above to complete parts (a) through (c).
a. Verify that the requirements for constructing a confidence interval about ˆpare satisfied.
b. Construct a 90% confidence interval for the population proportion of drug users who will report a headache as a side effect.
c. Interpret the confidence interval.
12.50 Solar Energy. America has been working towards harnessing the power of the sun to help supplement our dependence on fossil fuels since the early 20th century. AHarris Pollof 2,205 U.S. adults was conducted through an online survey between October 15 and 20, 2014.
Of those surveyed, 16% expected that in the following 2 to 5 years, solar energy will make hardly any contribution to meet their daily energy needs. Find and interpret a 90% confidence interval for the proportion of all U.S. adults who believed that solar energy will hardly contribute to meeting our energy needs in the succeeding 2 to 5 years.
12.51 Asthmatics and Sulfites. In the article “Explaining an Unusual Allergy,” appearing on the Everyday Health Network, Dr. A. Feldweg explained that allergy to sulfites is usually seen in patients with asthma. The typical reaction is a sudden increase in asthma symptoms after eating a food containing sulfites. Studies are performed to estimate the percentage of the nation’s 10 million asth- matics who are allergic to sulfites. In one survey, 38 of 500 randomly selected U.S. asthmatics were found to be allergic to sulfites. Find and interpret a 95% confidence interval for the proportion,p, of all U.S. asthmatics who are allergic to sulfites.
12.52 Christmas Drink. In a 2014 nationwide Harris Pollsur- vey about beverage preferences, 1951 American adults were asked about their preferred beverages during different holidays throughout the year. Overall, 47% of the adults voted for table wine as their pre- ferred beverage for Christmas. Determine and interpret a 95% confi- dence interval for the proportion,p, of all American adults who prefer wine during holidays.
12.53 Factory Farming Funk. During one year, theU.S. Environ- mental Protection Agencyreported that concentrated animal feeding
12.1 Confidence Intervals for One Population Proportion 577 operations (CAFOs) dump 2 trillion pounds of waste into the envi-
ronment annually, contaminating the ground water in 17 states and polluting more than 35,000 miles of our nation’s rivers. In a survey of 1000 registered voters bySnell, Perry and Associates, 80% favored the creation of standards to limit such pollution and, in general, viewed CAFOs unfavorably. Find and interpret a 99% confidence interval for the percentage of all registered voters who favor the crea- tion of standards on CAFO pollution and, in general, view CAFOs unfavorably.
12.54 The Nipah Virus. During one year, Malaysia was the site of an encephalitis outbreak caused by the Nipah virus, a paramyxovirus that appears to spread from pigs to workers on pig farms. As reported by K. Goh et al. in the paper “Clinical Features of Nipah Virus Encephalitis among Pig Farmers in Malaysia” (New England Journal of Medicine, Vol. 342, No. 17, pp. 1229–1235), neurologists from the University of Malaysiafound that, among 94 patients infected with the Nipah virus, 30 died from encephalitis. Find and interpret a 90%
confidence interval for the percentage of Malaysians infected with the Nipah virus who would die from encephalitis.
12.55 Literate Adults. Suppose that you have been hired to esti- mate the percentage of adults in your state who are literate. You take a random sample of 100 adults and find that 96 are literate. You then obtain a 95% confidence interval of
0.96±1.96ã
(0.96)(0.04)/100,
or 0.922 to 0.998. From it you conclude that you can be 95% confi- dent that the percentage of all adults in your state who are literate is somewhere between 92.2% and 99.8%. Is anything wrong with this reasoning?
12.56 IMR in Singapore. The infant mortality rate (IMR) is the number of infant deaths per 1000 live births. Suppose that you have been commissioned to estimate the IMR in Singapore. From a ran- dom sample of 1109 live births in Singapore, you find that 0.361%
of them resulted in infant deaths. You next find a 90% confidence interval:
0.00361±1.645ã
(0.00361)(0.99639)/1109,
or 0.000647 to 0.00657. You then conclude, “I can be 90% confident that the IMR in Singapore is somewhere between 0.647 and 6.57.”
How did you do?
12.57 Bank Breakup. In a nationwide survey, conducted byPulse Opinion Research, LLCforRasmussen Reports, a sample of Amer- ican adults were asked whether they favor a plan to break up the 12 megabanks, which currently control about 69% of the banking in- dustry; 50% of those sampled responded in the affirmative. Accord- ing to the report, “the margin of sampling error is+/−3 percentage points with a 95% level of confidence.” Find and interpret a 95% con- fidence interval for the percentage of all American adults who favor a plan to break up the 12 megabanks.
12.58 Online Tax Returns. According to theInternal Revenue Ser- vice, among people entitled to tax refunds, those who file online re- ceive their refunds twice as fast as paper filers. A study conducted by International Communications Research(ICR) of Media, Pennsylva- nia, found that 57% of those polled said that they are not worried about the privacy of their financial information when filing their tax returns online. The survey had a margin of error of plus or minus 3 percentage points (for a 0.95 confidence level). Use this informa- tion to determine a 95% confidence interval for the percentage of all people who are not worried about the privacy of their financial infor- mation when filing their tax returns online.
12.59 Asthmatics and Sulfites. Refer to Exercise 12.51.
a. Determine the margin of error for the estimate ofp.
b. Obtain a sample size that will ensure a margin of error of at most 0.01 for a 95% confidence interval without making a guess for the observed value of ˆp.
c. Find a 95% confidence interval forpif, for a sample of the size determined in part (b), the proportion of asthmatics sampled who are allergic to sulfites is 0.071.
d. Determine the margin of error for the estimate in part (c) and com- pare it to the margin of error specified in part (b).
e. Repeat parts (b)–(d) if you can reasonably presume that the pro- portion of asthmatics sampled who are allergic to sulfites will be at most 0.10.
f. Compare the results you obtained in parts (b)–(d) with those obtained in part (e).
12.60 Christmas Drink. Refer to Exercise 12.52.
a. Find the margin of error for the estimate of p.
b. Obtain a sample size that will ensure a margin of error of at most 0.02 for a 95% confidence interval without making a guess for the observed value of ˇS.
c. Find a 95% confidence interval forpif, for a sample of the size determined in part (b), 50.0% of those sampled drink wine as their preferred holiday beverage.
d. Determine the margin of error for the estimate in part (c) and com- pare it to the margin of error specified in part (b).
e. Repeat parts (b)–(d) if you can reasonably presume that the per- centage of adults sampled who drink wine as their preferred holi- day beverage will be at most 55%.
f. Compare the results you obtained in parts (b)–(d) with those obtained in part (e).
12.61 Factory Farming Funk. Refer to Exercise 12.53.
a. Find the margin of error for the estimate of the percentage.
b. Obtain a sample size that will ensure a margin of error of at most 1.5 percentage points for a 99% confidence interval without mak- ing a guess for the observed value of ˆp.
c. Find a 99% confidence interval forpif, for a sample of the size determined in part (b), 82.2% of the registered voters sampled favor the creation of standards on CAFO pollution and, in gen- eral, view CAFOs unfavorably.
d. Determine the margin of error for the estimate in part (c) and com- pare it to the margin of error specified in part (b).
e. Repeat parts (b)–(d) if you can reasonably presume that the per- centage of registered voters sampled who favor the creation of standards on CAFO pollution and, in general, view CAFOs un- favorably will be between 75% and 85%.
f. Compare the results you obtained in parts (b)–(d) with those ob- tained in part (e).
12.62 The Nipah Virus. Refer to Exercise 12.54.
a. Find the margin of error for the estimate of the percentage.
b. Obtain a sample size that will ensure a margin of error of at most 5 percentage points for a 90% confidence interval without making a guess for the observed value of ˆp.
c. Find a 90% confidence interval forpif, for a sample of the size determined in part (b), 28.8% of the sampled Malaysians infected with the Nipah virus die from encephalitis.
d. Find the margin of error for the estimate in part (c) and compare it to the margin of error specified in part (b).
e. Repeat parts (b)–(d) if you can reasonably presume that the percentage of sampled Malaysians infected with the Nipah virus who would die from encephalitis would be between 25%
and 40%.
f. Compare the results you obtained in parts (b)–(d) with those obtained in part (e).
12.63 Product Response Rate. A company manufactures goods that are sold exclusively by mail order. The director of market re- search needed to test market a new product. She planned to send brochures to a random sample of households and use the propor- tion of orders obtained as an estimate of the true proportion, known as the product response rate. The results of the market research were to be utilized as a primary source for advance production planning, so the director wanted the figures she presented to be as accurate as possible. Specifically, she wanted to be 95% confident that the estimate of the product response rate would be accurate to within 1%.
a. Without making any assumptions, determine the sample size required.
b. Historically, product response rates for products sold by this com- pany have ranged from 0.5% to 4.9%. If the director had been willing to assume that the sample product response rate for this product would also fall in that range, find the required sample size.
c. Compare the results from parts (a) and (b).
d. Discuss the possible consequences if the assumption made in part (b) turns out to be incorrect.
12.64 Indicted Governor. On Thursday, June 13, 1996, then- Arizona Governor Fife Symington was indicted on 23 counts of fraud and extortion. Just hours after the federal prosecutors announced the indictment, several polls were conducted of Arizonans asking whether they thought Symington should resign. A poll conducted byResearch Resources, Inc., that appeared in thePhoenix Gazette, revealed that 58% of Arizonans felt that Symington should resign; it had a margin of error of plus or minus 4.9 percentage points. Another poll, conducted by Phoenix-based Behavior Research Center and appearing in theTempe Daily News, reported that 54% of Arizonans felt that Symington should resign; it had a margin of error of plus or minus 4.4 percentage points. Can the conclusions of both polls be correct? Explain your answer.
12.65 President’s Job Rating. A poll conducted by Gallup in December 2013 asked a sample of American adults whether they approved of the way President Obama was doing his job; 42% said yes, with a margin of error of plus or minus 3 percentage points.
During that same time period,Quinnipiac Universityasked the same question of a sample of American adults; 38% said yes, with a margin of error of plus or minus 2 percentage points. Can the conclusions of both polls be correct? Explain your answer.
Extending the Concepts and Skills
12.66 What important theorem in statistics implies that, for a large sample size, the possible sample proportions of that size have approx- imately a normal distribution?
12.67 In discussing the sample size required for obtaining a confi- dence interval with a prescribed confidence level and margin of error, we made the following statement: “If we have in mind a likely range for the observed value of ˆp, then, in light of Fig. 12.1, we should take as our educated guess for ˆpthe value in the range closest to 0.5.”
Explain why.
12.68 In discussing the sample size required for obtaining a confi- dence interval with a prescribed confidence level and margin of error, we made the following statement: “. . . we should be aware that, if the observed value of ˆpis closer to 0.5 than is our educated guess, the margin of error will be larger than desired.” Explain why.
One-Proportion Plus-Fourz-Interval Procedure. To obtain a plus- fourz-interval for a population proportion, we first add two successes and two failures to our data (hence, the term “plus four”) and then apply Procedure 12.1 on page 570 to the new data. In other words, in place of ˆp(which isx/n), we use ˜p=(x+2)/(n+4). Conse- quently, for a confidence level of 1−α, the endpoints of the plus-four z-interval are
˜
p±zα/2ã
˜
p(1−p)˜ /(n+4).
As a rule of thumb, the one-proportion plus-fourz-interval procedure should be used only with confidence levels of 90% or greater and sample sizes of 10 or more.
In each of Exercises12.69–12.74, we have given the number of suc- cesses and the sample size for a simple random sample from a popu- lation. In each case,
a. use the one-proportion plus-four z-interval procedure to find the required confidence interval.
b. compare your result with the corresponding confidence interval found in Exercises 12.25–12.30, if finding such a confidence in- terval was appropriate.
12.69 x=8,n=40, 95% level 12.70 x=10,n=40, 90% level 12.71 x=35,n=50, 99% level 12.72 x=40,n=50, 95% level 12.73 x=16,n=20, 90% level 12.74 x=3,n=100, 99% level
In each of Exercises12.75–12.78, use the one-proportion plus-four z-interval procedure to find the required confidence interval. Inter- pret your results.
12.75 Working with Millions. A poll byGallupasked, “If you won 10 million dollars in the lottery, would you continue to work or stop working?” Of the 1039 American adults surveyed, 707 said that they would continue working. Obtain a 95% confidence interval for the proportion of all American adults who would continue working if they won 10 million dollars in the lottery.
12.76 Federal Income Tax. A telephonic poll regarding the amount of federal income tax paid was conducted. From a sample of 1026 adults of 30–45 years, 534 said that the amount was too high.
Determine a 95% confidence interval for the percentage of all adults aged 30–45 years, who felt the federal income tax they pay is high.
12.77 Breast-Feeding. In aNew York Timesarticle “More Mothers Breast-Feed, in First Months at Least,” G. Harris reported that 77%
of new mothers breast-feed their infants at least briefly, the highest rate seen in the United States in more than a decade. His report was based on data for 434 infants from theNational Health and Nutrition Examination Survey, which involved in-person interviews and phys- ical examinations. Find a 90% confidence interval for the percentage of all new mothers who breast-feed their infants at least briefly.
12.78 Offshore Drilling. In the February 2013 article “Offshore Drilling Support High as Deepwater Horizon Oil Spill Trial Opens,”
E. Swanson reported on a HuffPost and YouGov poll that asked Americans what they think about increased offshore drilling for oil and natural gas. Of the 1000 U.S. adults surveyed, 280 said that they were opposed. Find a 99% confidence interval for the proportion of all U.S. adults who, at the time, opposed increased offshore drilling for oil and natural gas.