Stats Data and Models

Richard D. De Veaux, Paul F. Velleman, David E. Bock

Chapter 18

Confidence Intervals for Proportions - all with Video Answers

Educators

Chapter Questions

Problem 1

Pew Research, in November $2011,$ polled a random sample of 799 U.S. teens about Internet use. $49 \%$ of those teens reported that they had misrepresented their age online to gain access to websites and online services.
a) Explain the meaning of $\hat{p}=0.49$ in the context of this situation.
b) Calculate the standard error of $\hat{p}$
c) Explain what this standard error means in the context of this situation.

James Kiss

James Kiss

Numerade Educator

Problem 2

Gallup regularly conducts a poll asking people to imagine a ladder with 10 rungs. Rung 0 represents the worst possible life and rung 10 represents the best possible life. Respondents are asked what rung they would say they are on. Responses are classified as "Thriving" (standing on rung 7 or higher, and expecting to be on rung 8 or higher five years from now), "Suffering" (standing on rung 4 or lower and expecting to be on rung 4 or lower 5 years from now), or "Struggling" (not thriving or suffering). The poll for November $22-25,2011,$ found $4 \%$ of respondents to be suffering. The polls all use a sample size of approximately 1500 people. (www.gallup .com/poll/152033/Fewer-Americans-Thriving-2011-
2010.aspx)
a) Explain the meaning of $" \hat{p}=0.04$ " in the context of this situation.
b) Calculate the standard error of $\hat{p}$.
c) Explain what this standard error means in the context of this situation.

James Kiss

Numerade Educator

Problem 3

The $95 \%$ confidence interval for the number of teens in Exercise 1 who reported that they had misrepresented their age online is from $45.6 \%$ to $52.5 \%$
a) Interpret the interval in this context.
b) Explain the meaning of "95\% confident" in this context.

James Kiss

Numerade Educator

Problem 4

In a 2010 Pew Research study on trends in marriage and family, $4 \%$ of randomly selected respondents said that the trend of more single women having children is "a good thing." The $95 \%$ confidence interval is from $2.9 \%$ to $5.1 \%(n=1229)$.
a) Interpret this interval in this context.
b) Explain the meaning of "95\% confident" in this context.

James Kiss

Numerade Educator

Problem 5

St. Norbert's College in Green Bay, Wisconsin, and Wisconsin Public Radio conduct an annual poll of Wisconsinites about political opinions. The Fall 2011 survey asked a random sample of 402 adult Wisconsin residents whether they think things in the country are going in the right direction or in the wrong direction. $66 \%$ said that things were going in the wrong direction.
a) Calculate the margin of error for the proportion of all adult Wisconsin residents who think things are going in the wrong direction for $90 \%$ confidence.
b) Would the margin of error be larger or smaller for $95 \%$ confidence? Explain.

James Kiss

Numerade Educator

Problem 6

In the Gallup poll described in Exercise $2,49 \%$ of those polled considered themselves to be "thriving."
a) Calculate the margin of error for the proportion of all American adults who were rated as "thriving" for $99 \%$ confidence.
b) Would the margin of error be larger or smaller for $95 \%$ confidence? Explain.

James Kiss

Numerade Educator

Problem 7

Consider the St. Norbert's College poll of Exercise 5
a) Are the assumptions and conditions met?
b) How many people would need to be surveyed for a $90 \%$ confidence interval to ensure the margin of error would be less than $2 \%$ ?

James Kiss

Numerade Educator

Problem 8

In Exercise $4,$ we saw that $4 \%$ of people think the trend of more single women having children is a "good thing."
a) Are the conditions for constructing a confidence interval met?
b) How many people would need to be surveyed for a $95 \%$ confidence interval to ensure the margin of error would be less than $3 \%$ ? (Hint: Don't use $\hat{p}=0.5$.)

James Kiss

Numerade Educator

Problem 9

A TV newscaster reports the results of a poll of voters, and then says, "The margin of error is plus or minus $8 \%$." Explain carefully what that means.

James Kiss

Numerade Educator

Problem 10

A medical researcher estimates the percentage of children exposed to lead-based paint, adding that he believes his estimate has a margin of error of about $3 \%$. Explain what the margin of error means.

James Kiss

Numerade Educator

Problem 11

For each situation described below, identify the population and the sample, explain what $p$ and $\hat{p}$ represent, and tell whether the methods of this chapter can be used to create a confidence interval.
a) Police set up an auto checkpoint at which drivers are stopped and their cars inspected for safety problems. They find that 16 of the 117 cars stopped have at least one safety violation. They want to estimate the percentage of all cars that may be unsafe.
b) An online blog puts a poll on its website asking readers to rate the quality of its articles. Of the 529 people who voted, 350 rated the articles very highly. The blog is interested in gaging its users' satisfaction.
c) A school is considering requiring students to wear uniforms. The PTA surveys parent opinion by sending a questionnaire home with all 973 students; 358 surveys are returned, with 103 families in favor of the change.
d) A college admits 1627 freshmen one year, and four years later, 1462 of them graduate on time. The college wants to estimate the percentage of all their freshman enrollees who graduate on time.

Kari Hasz

Numerade Educator

Problem 12

Consider each situation described. Identify the population and the sample, explain what $p$ and $\hat{p}$ represent, and tell whether the methods of this chapter can be used to create a confidence interval.
a) A consumer group hoping to assess customer experiences with auto dealers surveys 167 people who recently bought new cars; $3 \%$ of them expressed dissatisfaction with the salesperson.
b) What percent of college students have cell phones? 2883 students were asked as they entered a football stadium, and 2430 said they had phones with them.
c) Two hundred forty potato plants in a field in Maine are randomly checked, and only 7 show signs of blight. How severe is the blight problem for the U.S. potato industry?
d) Twelve of the 309 employees of a small company suffered an injury on the job last year. What can the company expect in future years?

Kari Hasz

Numerade Educator

Problem 13

A catalog sales company promises to deliver orders placed on the Internet within 3 days. Follow-up calls to a few randomly selected customers show that a $95 \%$ confidence interval for the proportion of all orders that arrive on time is $85 \% \pm 5 \%$. What does this mean? Are these conclusions correct? Explain.
a) Between $80 \%$ and $90 \%$ of all orders arrive on time.
b) Ninety-five percent of all random samples of customers will show that $85 \%$ of orders arrive on time.
c) One can be $95 \%$ confident that the true population proportion of orders that arrive within 3 days is between $80 \%$ and $90 \%$
d) We are $95 \%$ sure that between $80 \%$ and $90 \%$ of the orders placed by the sampled customers arrived on time.
e) There is a $95 \%$ chance that on any given day, a customer has an $80 \%$ to $90 \%$ chance of receiving his package on time.

James Kiss

Numerade Educator

Problem 14

In January $2002,$ two students made worldwide headlines by spinning a Belgian euro 250 times and getting 140 heads $-$ that's $56 \%$. That makes the $90 \%$ confidence interval $(51 \%, 61 \%) .$ What does this mean? Are these conclusions correct? Explain.
a) Between $51 \%$ and $61 \%$ of all euros are unfair.
b) We are $90 \%$ sure that in this experiment this euro landed heads on between $51 \%$ and $61 \%$ of the spins.
c) We are $90 \%$ sure that spun euros will land heads between $51 \%$ and $61 \%$ of the time.
d) If you spin a euro many times, you can be $90 \%$ sure of getting between $51 \%$ and $61 \%$ heads.
e) Ninety percent of all spun euros will land heads between $51 \%$ and $61 \%$ of the time.

James Kiss

Numerade Educator

Problem 15

Several factors are involved in the creation of a confidence interval. Among them are the sample size, the level of confidence, and the margin of error. Which statements are true?
a) For a given sample size, higher confidence means a larger margin of error.
b) For a specified confidence level, smaller samples provide smaller margins of error.
c) For a fixed margin of error, larger samples provide greater confidence.
d) For a given confidence level, halving the margin of error requires a sample twice as large.

Robin Corrigan

Numerade Educator

Problem 16

Several factors are involved in the creation of a confidence interval. Among them are the sample size, the level of confidence, and the margin of error. Which statements are true?
a) For a given sample size, reducing the margin of error will mean lower confidence.
b) For a certain confidence level, you can get a smaller margin of error by selecting a bigger sample.
c) For a fixed margin of error, smaller samples will mean lower confidence.
d) For a given confidence level, a sample 9 times as large will make a margin of error one third as big.

Robin Corrigan

Numerade Educator

Problem 17

What fraction of cars is made in Japan? The computer output below summarizes the results of a random sample of 50 autos. Explain carefully what it tells you.
z-Interval for proportion With $95.00 \%$ confidence,
$$
0.32797941<\mathrm{P}(\text { japan })<0.50820747
$$

James Kiss

Numerade Educator

Problem 18

A study of 902 decisions (to grant parole or not) made by the Nebraska Board of Parole produced the following computer output. Assuming these cases are representative of all cases that may come before the Board, what can you conclude?
z-Interval for proportion With $95.00 \%$ confidence,
$$
0.56100658<\mathrm{P}(\text { parole })<0.62524619
$$

James Kiss

Numerade Educator

Problem 19

In December 2011, Consumer Reports published their study of labeling of seafood sold in New York, New Jersey, and Connecticut. They purchased 190 pieces of seafood from various kinds of food stores and restaurants in the three states and genetically compared the pieces to standard gene fragments that can identify the species. Laboratory results indicated that $22 \%$ of these packages of seafood were mislabeled, incompletely labeled, or misidentified by store or restaurant employees.
a) Construct a $95 \%$ confidence interval for the proportion of all seafood packages in those three states that are mislabeled or misidentified.
b) Explain what your confidence interval says about seafood sold in these three states.
c) A 2009 report by the Government Accountability Board says that the Food and Drug Administration has spent very little time recently looking for seafood fraud. Suppose an official said, "That's only 190 packages out of the billions of pieces of seafood sold in a year. With the small number tested, I don't know that one would want to change one's buying habits." (An official was quoted similarly in a different but similar context). Is this argument valid? Explain.

James Kiss

Numerade Educator

Problem 20

The December 2011 Consumer Reports study described in Exercise 19 also found that 12 of the 22 "red snapper" packages tested were a different kind of fish.
a) Are the conditions for creating a confidence interval satisfied? Explain.
b) Construct a $95 \%$ confidence interval.
c) Explain what your confidence interval says about "red snapper" sold in these three states.

James Kiss

Numerade Educator

Problem 21

In a poll taken in December 2012, Gallup asked 1006 national adults whether they were baseball fans; $48 \%$ said they were. Three years earlier, in February 2008 , only $35 \%$ of a similar-size sample had reported being baseball fans.
a) Find the margin of error for the 2012 poll if we want $90 \%$ confidence in our estimate of the percent of national adults who are baseball fans.
b) Explain what that margin of error means.
c) If we wanted to be $99 \%$ confident, would the margin of error be larger or smaller? Explain.
d) Find that margin of error.
e) In general, if all other aspects of the situation remain the same, will smaller margins of error produce greater or less confidence in the interval?

Kari Hasz

Numerade Educator

Problem 22

The Pew Research poll described in Exercise 1 found that $49 \%$ of a sample of 799 teens admitted to misrepresenting their age online to access websites and online services. (Treat this as a simple random sample.)
a) Find the margin of error for this poll if we want $95 \%$ confidence in our estimate of the percent of American teens who have misrepresented their age online.
b) Explain what that margin of error means.
c) If we only need to be $90 \%$ confident, will the margin of error be larger or smaller? Explain.
d) Find that margin of error.
e) In general, if all other aspects of the situation remain the same, would smaller samples produce smaller or larger margins of error?

Kari Hasz

Numerade Educator

Problem 23

The Paralyzed Veterans of America is a philanthropic organization that relies on contributions. They send free mailing labels and greeting cards to potential donors on their list and ask for a voluntary contribution. To test a new campaign, they recently sent letters to a random sample of 100,000 potential donors and received 4781 donations.
a) Give a $95 \%$ confidence interval for the true proportion of their entire mailing list who may donate.
b) A staff member thinks that the true rate is $5 \%$. Given the confidence interval you found, do you find that percentage plausible?

James Kiss

Numerade Educator

Problem 24

First USA, a major credit card company, is planning a new offer for their current cardholders. The offer will give double airline miles on purchases for the next 6 months if the cardholder goes online and registers for the offer. To test the effectiveness of the campaign, First USA recently sent out offers to a random sample of 50,000 cardholders. Of those, 1184 registered.
a) Give a $95 \%$ confidence interval for the true proportion of those cardholders who will register for the offer.
b) If the acceptance rate is only $2 \%$ or less, the campaign won't be worth the expense. Given the confidence interval you found, what would you say?

James Kiss

Numerade Educator

Problem 25

An insurance company checks police records on 567 accidents selected at random and notes that teenagers were at the wheel in 86 of them.
a) Create a $95 \%$ confidence interval for the percentage of all auto accidents that involve teenage drivers.
b) Explain what your interval means.
c) Explain what "95\% confidence" means.
d) A politician urging tighter restrictions on drivers' licenses issued to teens says, "In one of every five auto accidents, a teenager is behind the wheel." Does your confidence interval support or contradict this statement? Explain.

Kari Hasz

Numerade Educator

Problem 26

Direct mail advertisers send solicitations (a.k.a. "junk mail") to thousands of potential customers in the hope that some will buy the company's product. The acceptance rate is usually quite low. Suppose a company wants to test the response to a new flyer, and sends it to 1160 people randomly selected from their mailing list of over 200,000 people. They get orders from 101 of the recipients.
a) Create a $90 \%$ confidence interval for the percentage of people the company contacts who may buy something.
b) Explain what this interval means.
c) Explain what "90\% confidence" means.
d) The company must decide whether to now do a mass mailing. The mailing won't be cost-effective unless it produces at least a $5 \%$ return. What does your confidence interval suggest? Explain.

Kari Hasz

Numerade Educator

Problem 27

Some food retailers propose subjecting food to a low level of radiation in order to improve safety, but sale of such "irradiated" food is opposed by many people. Suppose a grocer wants to find out what his customers think. He has cashiers distribute surveys at checkout and ask customers to fill them out and drop them in a box near the front door. He gets responses from 144 customers, of whom 74 oppose the radiation treatments. What can the grocer conclude about the opinions of all his customers?

James Kiss

Numerade Educator

Problem 28

The mayor of a small city has suggested that the state locate a new prison there, arguing that the construction project and resulting jobs will be good for the local economy. A total of 183 residents show up for a public hearing on the proposal, and a show of hands finds only 31 in favor of the prison project. What can the city council conclude about public support for the mayor's initiative?

Prashant Bana

Numerade Educator

Problem 29

In the survey on the death penalty you read about in the chapter, the Gallup Poll actually split the sample at random, asking 510 respondents the question quoted earlier, "Generally speaking, do you believe the death penalty is applied fairly or unfairly in this country today?" The other 510 were asked, "Generally speaking, do you believe the death penalty is applied unfairly or fairly in this country today?" Seems like the same question, but sometimes the order of the choices matters. Suppose that for the second way of phrasing it, $64 \%$ said they thought the death penalty was fairly applied. (Recall that $58 \%$ of the original 510 thought the same thing.)
a) What kind of bias may be present here?
b) If we combine them, considering the overall group to be one larger random sample of 1020 respondents, what is a $95 \%$ confidence interval for the proportion of the general public that thinks the death penalty is being fairly applied?
c) How does the margin of error based on this pooled sample compare with the margins of error from the separate groups? Why?

James Kiss

Numerade Educator

Problem 30

A city ballot includes a local initiative that would legalize gambling. The issue is hotly contested, and two groups decide to conduct polls to predict the outcome. The local newspaper finds that $53 \%$ of 1200 randomly selected voters plan to vote "yes," while a college Statistics class finds $54 \%$ of 450 randomly selected voters in support. Both groups will create $95 \%$ confidence intervals.
a) Without finding the confidence intervals, explain which one will have the larger margin of error.
b) Find both confidence intervals.
c) Which group concludes that the outcome is too close to call? Why?

Prashant Bana

Numerade Educator

Problem 31

Vitamin D, whether ingested as a dietary supplement or produced naturally when sunlight falls on the skin, is essential for strong, healthy bones. The bone disease rickets was largely eliminated in England during the $1950 \mathrm{~s}$, but now there is concern that a generation of children more likely to watch TV or play computer games than spend time outdoors is at increased risk. A recent study of 2500 children randomly selected from all parts of England found $20 \%$ of them deficient in vitamin D.
a) Find a $98 \%$ confidence interval.
b) Explain carefully what your interval means.
c) Explain what "98\% confidence" means.

James Kiss

Numerade Educator

Problem 32

A 2011 Gallup poll found that $76 \%$ of Americans believe that high achieving high school students should be recruited to become teachers. This poll was based on a random sample of 1002 Americans.
a) Find a $90 \%$ confidence interval for the proportion of Americans who would agree with this.
b) Interpret your interval in this context.
c) Explain what "90\% confidence" means.
d) Do these data refute a pundit's claim that $2 / 3$ of Americans believe this statement? Explain.

James Kiss

Numerade Educator

Problem 33

In January 2014 AP-GfK polled 1060 U.S. adults to find if people were more concerned with privacy or security. Privacy concerns outweighed concerns about being safe from terrorists for 646 out the 1060 polled. Of the 1060 adults about 180 are 65 and older whose concerns may be different from the rest of the population.
a) Do you expect the $95 \%$ confidence interval for the true proportion of those 65 and over who are more concerned with privacy to be wider or narrower than the $95 \%$ confidence interval for the true proportion of all U.S. consumers? Explain.
b) If $13 \%$ of the 1060 polled were $18-24$ years old, would you expect the margin of error for this group to be larger or smaller than for the 65 and older group? Explain.

James Kiss

Numerade Educator

Problem 34

ACT, Inc. reported that $74 \%$ of 1644 randomly selected college freshmen returned to college the next year. The study was stratified by type of college-public or private. The retention rates were $71.9 \%$ among 505 students enrolled in public colleges and $74.9 \%$ among 1139 students enrolled in private colleges.
a) Will the $95 \%$ confidence interval for the true national retention rate in private colleges be wider or narrower than the $95 \%$ confidence interval for the retention rate in public colleges? Explain.
b) Do you expect the margin of error for the overall retention rate to be larger or smaller? Explain.

James Kiss

Numerade Educator

Problem 35

Wildlife biologists inspect 157 deer taken by hunters and find 33 of them carrying ticks that test positive for Lyme disease.
a) Create a $90 \%$ confidence interval for the percentage of deer that may carry such ticks.
b) If the scientists want to cut the margin of error in half, how many deer must they inspect?
c) What concerns do you have about this sample?

James Kiss

Numerade Educator

Problem 36

Suppose ACT, Inc. wants to update their information from Exercise 34 on the percentage of freshmen that return for a second year of college.
a) They want to cut the stated margin of error in half. How many college freshmen must be surveyed?
b) Do you have any concerns about this sample? Explain.

James Kiss

Numerade Educator

Problem 37

It's believed that as many as $21 \%$ of adults over 50 never graduated from high school. We wish to see if this percentage is the same among the 25 to 30 age group.
a) How many of this younger age group must we survey in order to estimate the proportion of non-grads to within $10 \%$ with $90 \%$ confidence?
b) Suppose we want to cut the margin of error to $4 \%$. What's the necessary sample size?
c) What sample size would produce a margin of error of 5\%?

Kari Hasz

Numerade Educator

Problem 38

In preparing a report on the economy, we need to estimate the percentage of businesses that plan to hire additional employees in the next 60 days.
a) How many randomly selected employers must we contact in order to create an estimate in which we are $98 \%$ confident with a margin of error of $5 \%$ ?
b) Suppose we want to reduce the margin of error to $3 \%$. What sample size will suffice?
c) Why might it not be worth the effort to try to get an interval with a margin of error of only $1 \%$ ?

Kari Hasz

Numerade Educator

Problem 39

As in Exercise $37,$ we hope to estimate the percentage of adults aged 25 to 30 who never graduated from high school. What sample size would allow us to increase our confidence level to $95 \%$ while reducing the margin of error to only $2 \% ?$

James Kiss

Numerade Educator

Problem 40

Editors of the business report in Exercise 38 are willing to accept a margin of error of $4 \%$ but want $99 \%$ confidence. How many randomly selected employers will they need to contact?

James Kiss

Numerade Educator

Problem 41

A state's environmental agency worries that many cars may be violating clean air emissions standards. The agency hopes to check a sample of vehicles in order to estimate that percentage with a margin of error of $2 \%$ and $99 \%$ confidence. To gauge the size of the problem, the agency first picks 60 cars and finds 6 with faulty emissions systems. How many should be sampled for a full investigation?

James Kiss

Numerade Educator

Problem 42

During routine screening, a doctor notices that $22 \%$ of her adult patients show higher than normal levels of glucose in their blood - a possible warning signal for diabetes. Hearing this, some medical researchers decide to conduct a large-scale study, hoping to estimate the proportion to within $4 \%$ with $98 \%$ confidence. How many randomly selected adults must they test?

James Kiss

Numerade Educator

Problem 43

A newspaper reports that the governor's approval rating stands at $60 \%$. The article adds that the poll is based on a random sample of 1775 adults and has a margin of error of $3 \%$. What level of confidence did the pollsters use?

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 44

A TV news reporter says that a proposed constitutional amendment is likely to win approval in the upcoming election because a poll of 1505 likely voters indicated that $52 \%$ would vote in favor. The reporter goes on to say that the margin of error for this poll was $3 \%$.
a) Explain why the poll is actually inconclusive.
b) What confidence level did the pollsters use?

James Kiss

Numerade Educator