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Statistics The Art And Science Of Learning From Data 3rd Edition Alan Agresti, Christine A. Franklin - Solutions

An exam consists of 50 multiplechoice questions. Based on how much you studied, for any given question you think you have a probability of p = 0.70 of getting the correct answer. Consider the sampling distribution of the sample proportion of the 50 questions on which you get the correct answer. a.
According to a Boston Globe story, only about 1 in 6 Americans have blue eyes, whereas in 1900 about half had blue eyes. (Source: Data from The Boston Globe, October 17, 2006.) a. For a random sample of 100 living Americans, find the mean and standard deviation of the proportion that have blue
According to the Alzheimer’s Association1, as of 2011 Alzheimer’s disease affects 1 in 8 Americans over the age of 65. A study is planned of health problems faced by the elderly. For a random sample of Americans over the age of 65, report the shape, mean, and standard deviation of the sampling
In college basketball, a shot made from beyond a designated arc radiating about 20 feet from the basket is worth three points, instead of the usual two points given for shots made inside that arc. Over his career, University of Florida basketball player Lee Humphrey made 45% of his three-point
Suppose a baseball player has a 0.200 probability of getting a hit in each time at-bat. a. Describe the shape, mean, and standard deviation of the sampling distribution of the proportion of times the player gets a hit after 36 at-bats. b. Explain why it would not be surprising if the player has a
CNN conducted an exit poll of 1751 voters in the 2010 Senatorial election in New York between Charles Schumer and Jay Townsend. It is possible that all 1751 voters sampled happened to be Charles Schumer supporters. Investigate how surprising this would be, if actually 65% of the population voted
In Example 5 about Aunt Erma’s Restaurant, the daily sales follow a probability distribution that has a mean of μ = +900 and a standard deviation of σ = +300. This past week the daily sales for the seven days had a mean of $980 and a standard deviation of $276. a. Identify the mean and standard
Based on data from the 2010 major league baseball season, X = number of home runs the San Francisco Giants hits in a game has a mean of 1.0 and a standard deviation of 1.0. a. Do you think that X has a normal distribution? Why or why not? b. Suppose that this year X has the same distribution.
The 2006 AAPA survey of the population of physicians’ assistants who were working full time reported a mean annual income of $84,396 and standard deviation of $21,975. (Source: Data from 2006 AAPA survey [www.aapa.org].) a. Suppose the AAPA had randomly sampled 100 physicians’ assistants
For the population of individuals who own an iPhone, suppose p = 0.25 is the proportion that has a given app. For a particular iPhone owner, let x = 1 if they have the app and x = 0 otherwise. For a random sample of 50 people who have an iPhone: a. State the population distribution (that is, the
The scores on the Psychomotor Development Index (PDI), a scale of infant development, have a normal population distribution with mean 100 and standard deviation 15. An infant is selected at random. a. Find the z -score for a PDI value of 90. b. A study uses a random sample of 225 infants. Using the
According to recent General Social Surveys, in the United States the population distribution for adults of X = number of sex partners in the past 12 months has a mean of about 1.0 and a standard deviation of about 1.0. a. Does X have a normal distribution? Explain. b. For a random sample of 100
In many industrial production processes, measurements are made periodically on critical characteristics to ensure that the process is operating properly. Observations vary from item to item being produced, perhaps reflecting variability in material used in the process and/or variability in the way
Refer to the previous exercise. When a machine for dispensing a cola drink into bottles is in statistical control, the amount dispensed has a mean of 500 ml (milliliters) and a standard deviation of 4 ml. a. In constructing a control chart to monitor this process with periodic samples of size 4,
How would you explain to someone who has never studied statistics what a sampling distribution is? Explain by using the example of polls of 1000 Canadians for estimating the proportion who think the prime minister is doing a good job.
You’d like to estimate the proportion of all students in your school who are fluent in more than one language. You poll a random sample of 50 students and get a sample proportion of 0.12. Explain why the standard deviation of the sampling distribution of the sample proportion gives you useful
The owners of Aunt Erma’s Restaurant plan an advertising campaign with the claim that more people prefer the taste of their pizza (which we’ll denote by A) than the current leading fast-food chain selling pizza (which we’ll denote by D). To support their claim, they plan to randomly sample
Refer to Examples 1 and 2 about the exit poll, for which the sample size was 3889. In that election, 40.9% voted for Whitman. a. Define a binary random variable X taking values 0 and 1 that represents the vote for a particular voter (1 = vote for Whitman and 0 = another candidate). State its
Part b of Example 7 used the central limit theorem to approximate the probability of coming out ahead if you bet $10 on red on each of 40 different roulette wheel spins. For each spin, the winnings are $10 with probability 18/38 and - +10 with probability 20/38. You are interested in the
As the sample size increases, the standard deviation of the sampling distribution of x increases. Explain your answer.
Let X = GPA for students in your school. a. What would the sampling distribution of the sample mean look like if you sampled every student in the school, so the sample size equals the population size? b. How does the sampling distribution compare to the population distribution if we take a sample
The table provides the ages of all 50 heads of households in a small Nova Scotian fishing village. The distribution of these ages is characterized by Î¼ = 47.18 and Ïƒ = 14.74.a. Find the mean of the -values in part b. How does it compare to the value you would expect in a long run of repeated
For a single toss of a balanced coin, let x = 1 for a head and x = 0 for a tail. a. Construct the probability distribution for x, and calculate its mean. (You can think of this as the population distribution corresponding to a very long sequence of tosses.) b. The coin is flipped 10 times, yielding
Refer to the previous exercise. a. In part b, if the sampling distribution of the sample proportion had mean 0.409 and the standard deviation 0.008, give an interval of values within which the sample proportion will almost certainly fall. b. The sample proportion for Whitman from the exit poll was
Each student should bring 10 coins to class. For each coin, observe its age, the difference between the current year and the year on the coin. a. Using all the students’ observations, the class should construct a histogram of the sample ages. What is its shape? b. Now each student should find the
A baseball player in the major leagues who plays regularly will have about 500 atbats (that is, about 500 times he can be the hitter in a game) during a season. Suppose a player has a 0.300 probability of getting a hit in an at-bat. His batting average at the end of the season is the number of hits
Construct the sampling distribution of the sample proportion of heads obtained in the experiment of flipping a balanced coin: a. Once. (The possible sample proportion values are 0, if the flip is a tail, and 1, if the flip is a head. What are their probabilities?) b. Twice. (The possible samples
As part of a drug research study, individuals suffering from arthritis take an experimental pain relief medication. Suppose that 25% of all individuals who take the new drug experience a certain side effect. For a given individual, let X be either 1 or 0, depending on whether s/he experienced the
A study dealing with health care issues plans to take a sample survey of 1500 Americans to estimate the proportion who have health insurance and the mean dollar amount that Americans spent on health care this past year. a. Identify two population parameters that this study will estimate. b.
For the number of hours of TV watching, the 2008 GSS reported a mean of 2.98 for the 1324 white subjects, with a standard deviation of 2.66. The mean was 4.38 for the 188 black subjects, with a standard deviation of 3.58. Analyze these data, preparing a short report in which you mention the methods
Using data from the National Survey of Families and Households, a study (from S. South and G. Spitze, American Sociological Review, vol. 59, 1994, pp. 327€“347) reported the descriptive statistics in the following table for the hours spent on housework. Analyze these data. Summarize
When subjects in a recent GSS were asked whether they agreed with the following statements, the (yes, no) counts under various conditions were as follows: • Women should take care of running their homes and leave running the country up to men: (275, 1556). • It is better for everyone involved
An interval estimate for a mean is more informative than a point estimate, because with an interval estimate you can figure out the point estimate, but with the point estimate alone you have no idea how wide the interval estimate is. Explain why this statement is correct, illustrating using the
Why confidence intervals wider are when we use larger confidence levels but narrower when we use larger sample sizes, other things being equal?
To use the large sample confidence interval for p, you need at least 15 successes and 15 failures. Show that the smallest value of n for which the method can be used is (a) 30 when p̂ = 0.50. b) 50 when p̂ = 0.30. (c) 150 when p̂ = 0.10. That is, the overall n must increase as p̂ moves toward 0
Using the sample size formula n = [p̂ (1 - p̂)z2]/m2 for a proportion, explain the effect on n of (a) Increasing the confidence level and (b) Decreasing the margin of error.
A Gallup poll taken during June 2011 estimated that 8.8% of U.S. adults were unemployed. The poll was based on the responses of 30,000 U.S. adults in the workforce. Gallup reported that the margin of error associated with the poll is ±0.3 percentage points. Explain how they got this result.
A random sample of 50 records yields a 95% confidence interval of 21.5 to 23.0 years for the mean age at first marriage of women in a certain county. Explain what is wrong with each of the following interpretations of this interval. a. If random samples of 50 records were repeatedly selected, then
For many years, the General Social Survey has asked respondents whether they favor the death penalty for persons convicted of murder. Support has been quite high in the United States, one of few Western nations that currently has the death penalty. The following figure uses the 20 General Social
In 1994 (the most recent year asked), the General Social Survey asked, “During the last year, did anyone take something from you by using force—such as a stickup, mugging, or threat?” Of 1223 subjects, 31 answered yes and 1192 answered no. a. Find the point estimate of the proportion of the
You know the sample mean of n observations. Once you know (n - 1) of the observations, show that you can find the remaining one. In other words, for a given value of , the values of (n - 1) observations determine the remaining one. In summarizing scores on a quantitative variable, having (n - 1)
The large-sample confidence interval for a proportion substitutes p̂ for the unknown value of p in the exact standard error of p̂. A less approximate 95% confidence interval has endpoints determined by the p values that are 1.96 standard errors from the sample proportion, without estimating the
When the population distribution is normal, the population mean equals the population median. How good is the sample median as a point estimate of this common value? For a random sample, the estimated standard error of the sample median equals 1.25(s/√n). If the population is normal, explain why
To encourage subjects to make honest responses on sensitive questions, the method of randomized response is often used. Let€™s use your class to estimate the proportion that have had alcohol at a party. Before carrying out this method, the class should discuss what they would guess for the
When the 2000 GSS asked subjects (variable GRNSOL) if they would be willing to accept cuts in their standard of living to protect the environment, 344 of 1170 subjects said yes. a. Estimate the population proportion who would answer yes. b. Find the margin of error for a 95% confidence interval for
When the GSS recently asked subjects whether it should or should not be the government’s responsibility to impose strict laws to make industry do less damage to the environment (variable GRNLAWS), 1403 of 1497 subjects said yes. a. What assumptions are made to construct a 95% confidence interval
In the 2008 General Social Survey, respondents were asked if they favored or opposed the death penalty for people convicted of murder. Software shows resultsHere, X refers to the number of the respondents who were in favor. a. Show how to obtain the value reported under €œSample
Refer to the previous exercise. Show how you can get a 95% confidence interval for the proportion of American adults who were opposed to the death penalty from the confidence interval stated in the previous exercise for the proportion in favor.
A Harris poll of a random sample of 2113 adults in the United States in October 2010 reported that 72% of those polled believe that stem cell research has merit. (www.harrisinteractive.com/vault/Harris-Interactive-Poll-HealthDay-2010-10.pdf.) The results, presented using MINITAB software, areHere,
News coverage during a recent election projected that a certain candidate would receive 54.8% of all votes cast; the projection had a margin of error of {3% a. Give a point estimate for the proportion of all votes the candidate will receive. b. Give an interval estimate for the proportion of all
A recent survey of 1000 American women between the ages of 45 and 64 asked them what medical condition they most feared. Of those sampled, 61% said breast cancer, 8% said heart disease, and the rest picked other conditions. By contrast, currently about 3% of female deaths are due to breast cancer,
The 1996 GSS asked, “If the husband in a family wants children, but the wife decides that she does not want any children, is it all right for the wife to refuse to have children?” Of 699 respondents, 576 said yes. a. Find a 99% confidence interval for the population proportion who would say
A national television network takes an exit poll of 1400 voters after each has cast a vote in a state gubernatorial election. Of them, 660 say they voted for the Democratic candidate and 740 say they voted for the Republican candidate. a. Treating the sample as a random sample from the population
In the previous exercise, suppose the same proportions resulted from n = 140 (instead of 1400), with counts 66 and 74. a. Now does a 95% confidence interval allow you to predict the winner? Explain. b. Explain why the same proportions but with smaller samples provide less information.
Using the Confidence Interval for a Proportion applet on the text CD, let’s check that the large-sample confidence interval for a proportion may work poorly with small samples. Set n = 10 and p = 0.10. Generate 100 random samples, each of size 10, and for each one, form a 95% confidence interval
The 2008 General Social Survey asked, “What do you think is the ideal number of children for a family to have?” The 678 females who responded had a median of 2, mean of 3.22, and standard deviation of 1.99. a. What is the point estimate of the population mean? b. Find the standard error of the
Refer to the previous exercise. For the 604 males in the sample, the mean was 3.06 and the standard deviation was 1.92. a. Find the point estimate of the population mean, and show that its standard error is 0.078. b. The 95% confidence interval is 2.91 and 3.21. Explain what “95% confidence”
A study 6 compared various therapies for teenage girls suffering from anorexia, an eating disorder. For each girl, weight was measured before and after a fixed period of treatment. The variable measured was the change in weight, X = weight at the end of the study minus weight at the beginning of
From Example 7, trading volumes for General Electric stock on Mondays and Fridays duringFebruary through April of 2011 were given as followsThe Monday data have x = 51.82, s = 17.19, Q1 = 43, Median = 45, Q3 = 64. The Friday data have x = 50, s = 11.34, Q1 = 41.5, Median = 47, Q3 = 54.5. a.
Exercise 8.6 reported heights (in mm) of 55.5, 60.3, 60.6, 62.1, 65.5, and 69.2 for six seedlings fourteen days after germination.a. Using software or a calculator, verify that the 95% confidence interval for the population mean is (57.3, 67.1).b. Name two things you could do to get a narrower
Having estimated the mean amount of time spent watching TV in Example 6, we might want to estimate the mean for various groups, such as different religious groups. Let€™s consider Muslims. A recent GSS had responses on TV watching from seven subjects who identified their religion as
When the GSS asked in 2004, €œAbout how many hours per week do you spend sending and answering e-mail?€ the results were as shown in the TI €“ 83 + /84 screen shot. Explain how to interpret the information shown, and interpret the confidence interval at the 95%
For the question about e-mail in the previous exercise, suppose seven females in the GSS sample of age at least 80 had the responses 0, 0, 1, 2, 5, 7, 14, a. Using software or a calculator, find the sample mean and standard deviation and the standard error of the sample mean. b. Find and interpret
According to a union agreement, the mean income for all senior-level assembly-line workers in a large company equals $500 per week. A representative of a women’s group decides to analyze whether the mean income for female employees matches this norm. For a random sample of nine female employees,
Example 7 analyzed the trading volume of shares of General Electric stock between February and April 2011. Summary statistics of the data were calculated using MINITAB and are shown below:The 95% confidence intervals for the means are (40.2685, 63.3679) for Monday€™s volume and (42.7963,
For the FL Student Survey data file on the text CD, software reports the results for responses on the number of times a week the subject reads a newspaper:a. Is it plausible that Î¼ = 7, where Î¼ is the population mean for all Florida students? Explain. b. Suppose that the
The General Social Survey asks respondents to rate their political views on a seven-point scale, where 1 = extremely liberal, 4 = moderate, and 7 = extremely conservative. A researcher analyzing data from the 2008 GSS obtains MINITAB output:a. Show how to construct the confidence interval from the
One question (called NATFAREY) on the General Social Survey for the year 2008 asks, “Are we spending too much, too little, or about the right amount on assistance to the poor?” Of the 998 people who responded in 2008, 695 said too little, 217 said about right, and 86 said too much. a. Find the
A hospital administrator wants to estimate the mean length of stay for all inpatients using that hospital. Using a random sample of 100 records of patients for the previous year, she reports that “The sample mean was 5.3. In repeated random samples of this size, the sample mean could be expected
Find the margin of error for a 95% confidence interval for estimating the population mean when the sample standard deviation equals 100, with a sample size of (i) 400 and (ii) 1600. What is the effect of the sample size?
Find the margin of error for estimating the population mean when the sample standard deviation equals 100 for a sample size of 400, using confidence level (i) 95% and (ii) 99%. What is the effect of the choice of confidence level?
A company that sells its products through mail-order catalogs wants information about the success of its most recent catalog. The company decides to estimate the mean dollar amount of items ordered from those who received the catalog. For a random sample of 100 customers from their files, only 5
For the question, “How many children have you ever had?” use the GSS Website sda.berkeley.edu/GSS with the variable CHILDS to find the sample mean and standard deviation for the 2008 survey. a. Show how to obtain a standard error of 0.04 for a random sample of 2020 adults. b. Construct a 95%
Go to the Confidence Intervals for a Mean applet on the text CD. Choose the sample size of 50 and the skewed population distribution with μ = 100. Generate 100 random samples, each of size 50, and for each one form a 95% confidence interval for the mean.a. How many of the intervals fail to contain
The researcher planning the study in South Africa also will estimate the population proportion having at least a high school education. No information is available about its value. How large a sample size is needed to estimate it to within 0.07 with 95% confidence?
The Harvard study mentioned in the previous exercise estimated that 19% of college students abstain from drinking alcohol. To estimate this proportion in your school, how large a random sample would you need to estimate it to within 0.05 with probability 0.95, if before conducting the study a. You
A study is planned to estimate the proportion of businesses started in the year 2006 that had failed within five years of their start-up. How large a sample size is needed to guarantee estimating this proportion correct to within a. 0.10 with probability 0.95? b. 0.05 with probability 0.95? c. 0.05
In response to the GSS question in 2008 about the number of hours daily spent watching TV, the responses by the five subjects who identified themselves as Hindu were 3, 2, 1, 1, 1. a. Find a point estimate of the population mean for Hindus. b. The margin of error at the 95% confidence level for
An estimate is needed of the mean acreage of farms in Ontario, Canada. A 95% confidence interval should have a margin of error of 25 acres. A study 10 years ago in this province had a sample standard deviation of 200 acres for farm size. a. About how large a sample of farms is needed? b. A sample
How large a sample size do we need to estimate the mean annual income of Native Americans in Onondaga County, New York, correct to within $1000 with probability 0.99? No information is available to us about the standard deviation of their annual income. We guess that nearly all of the incomes fall
Explain the reasoning behind the following statement: “In studies about a very diverse population, large samples are often necessary, whereas for more homogeneous populations smaller samples are often adequate.” Illustrate for the problem of estimating mean income for all medical doctors in the
You randomly sample five students at your school to estimate the proportion of students who like tofu. All five students say they like it. a. Find the sample proportion who like it. b. Find the standard error. Does its usual interpretation make sense? c. Find a 95% confidence interval, using the
A pharmaceutical company proposes a new drug treatment for alleviating symptoms of PMS (premenstrual syndrome). In the first stages of a clinical trial, it was successful for 7 out of 10 women. a. Construct an appropriate 95% confidence interval for the population proportion. b. Is it plausible
A bank wants to estimate the proportion of people who would agree to take a credit card they offer if they send a particular mailing advertising it. For a trial mailing to a random sample of 100 potential customers, 0 people accept the offer. Can they conclude that fewer than 10% of their
Refer to Example 12 about weight readings of a scale. For 10 successive trials on the next day, the weight values wereExplain the steps of how you could use the bootstrap method to get a 95% confidence interval for a €œlong-run€ standard deviation of such values.
Researchers are interested in the effect of a certain nutrient on the growth rate of plant seedlings. Using a hydroponics grow procedure that utilized water containing the nutrient, they planted six tomato plants and recorded the heights of each plant 14 days after germination. Those heights,
We want a 95% confidence interval for the population proportion of students in a high school in Dallas, Texas, who can correctly find Iraq on an unlabeled globe. For a random sample of size 50, 10 get the correct answer.a. Using software or the Sampling Distributions applet on the text CD, set the
A July 2011 Gallup poll based on the responses of 1500 adults indicated that 46% of Americans approve of the job Barack Obama is doing as president. One way to summarize the findings of the poll is by saying, “It is estimated that 46% of Americans approve of the job Barack Obama is doing as
In February 2002, the Associated Press quoted a survey of 3000 British residents conducted by YouGov.com. It stated, “Only 21% wanted to see the monarchy abolished, but 53% felt it should become more democratic and approachable. No margin of error was given.” If the sample was random, find the
A poll of a random sample of n = 2000 Americans by the Pew Research Center (www.peoplepress.org ) indicated that 36% considered themselves “born-again” or evangelical Christians. How would you explain to someone who has not studied statistics: a. What it means to call this a point estimate. b.
The variable POSTLIFE in the 2008 General Social Survey asked, “Do you believe in life after death?” Of 1787 respondents, 1455 answered yes. A report based on these data stated that “81.4% of Americans believe in life after death. The margin of error for this result is plus or minus 1.85%.”
Refer to the previous exercise. The following printout shows results for the females in the sample, where X = the number answering yes. Explain how to interpret each item, in context.
Time magazine (July 15, 2002) quoted a poll of 10,000 Americans in which only 4% said they were vegetarians. a. What has to be assumed about this sample to construct a confidence interval for the population proportion of vegetarians? b. Construct a 99% confidence interval for the population
When a GSS asked 1326 subjects, “Do you believe in heaven?” (coded HEAVEN), the proportion who answered yes was 0.85. From results in the next section, the estimated standard deviation of this point estimate is 0.01. a. Find and interpret the margin of error for a 95% confidence interval for
Consider the statement that it is better for the man to work and the woman to tend the home, from the GSS (variable denoted FEFAM).a. Go to the Web site, sda.berkeley.edu/GSS. Find the number who agreed or strongly agreed with that statement and the sample size for the year 2008.b. Find the sample
The General Social Survey has asked respondents, “Do you think the use of marijuana should be made legal or not?” Go to the GSS Web site, sda.berkeley.edu/GSS. For the 2008 survey with variable GRASS: a. Of the respondents, how many said “legal” and how many said “not legal”? Report the
A report in 2004 by the U.S. National Center for Health Statistics provided an estimate of 20.4% for the percentage of Americans over the age of 18 who were currently smokers. The sample size was 30,000. Assuming that this sample has the characteristics of a random sample, a 99.9% confidence
Refer to the previous exercise. The same study provided the following results for estimating the proportion of adult Americans who have been lifetime abstainers from drinking alcohol.Explain how to interpret all results on this printout, in context.
When the GSS asked in 2004, €œAbout how many hours per week do you spend sending and answering e-mail?€ the eight males in the sample of age at least 75 responded:0, 1, 2, 2, 7, 10, 14, 15.a. The TI €“ 83 + /84 screen shot shows results of a statistical analysis for
As part of the 2000 census, the Census Bureau surveyed 700,000 households to study transportation to work. They reported that 76.3% drove alone to work, 11.2% carpooled, 5.1% took mass transit, 3.2% worked at home, 0.4% bicycled, and 3.8% took other means. a. With such a large survey, explain why

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