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

a. Show how the t -score for a 95% confidence interval changes as the sample size increases from 10 to 20 to 30 to infinity. b. What does the answer in part a suggest about how the t distribution compares to the standard normal distribution?
In a recent GSS, the responses about the number of hours daily spent watching TV for the five subjects who identified themselves as Buddhists were 0, 5, 0, 1, 2. a. Find the mean, standard deviation, and standard error. b. Construct a 95% confidence interval for the population mean. c. Specify the
In 2003, the American Psychological Association conducted a survey (at research.apa.org) of a random sample of psychologists to estimate mean incomes for psychologists with various academic degrees and levels of experience. Of the 31 psychologists who received a master’s degree in 2003, the mean
The GSS has asked “On how many days in the past seven days have you felt lonely?” At sda.berkeley.edu/GSS, enter LONELY as the variable, check Summary Statistics in the menu of table options, and click on Run the Table to see the responses.a. Report the percentage making each response and the
Refer to the previous exercise. Interpret each item on the following printout that software reports for psychologists with a doctorate but with less than one year of experience.
The General Social Survey has asked subjects, “How long have you lived in the city, town, or community where you live now?” The responses of 1415 subjects in one survey had a mode of less than 1 year, a median of 16 years, a mean of 20.3 and a standard deviation of 18.2. a. Do you think that
A recent GSS asked, “How many days in the past seven days have you felt sad?” The 816 women who responded had a median of 1, mean of 1.81, and standard deviation of 1.98. The 633 men who responded had a median of 1, mean of 1.42, and standard deviation of 1.83. a. Find a 95% confidence interval
Refer to the previous exercise. This question was asked of 10 students in a class at the University of Wisconsin recently. The responses were 0, 0, 1, 0, 7, 2, 1, 0, 0, 3. Find and interpret a 90% confidence interval for the population mean, and indicate what you would have to assume for this
The 1996 GSS asked, “How many days in the past seven days have you felt happy?” (This was the most recent year this question was posed.) a. Using the GSS variable HAPFEEL, verify that the sample had a mean of 5.27 and a standard deviation of 2.05. What was the sample size? b. Find the standard
Use the Mountain Bike data file from the text CD, shown also below. a. Form a 95% confidence interval for the population mean price of all mountain bikes. Interpret. b. What assumptions are made in forming the interval in part a? State at least one important assumption that does not seem to be
For eBay auctions of the iPad2 64GB 3G Wi-Fi units, a sample was taken in July 2011 where the Buy-it-Now prices were (in dollars): 1388, 1199, 1100, 1099, 1088, 1049, 1026, 999, 998, 978, 949, 930 a. Explain what a parameter might represent that you could estimate with these data. b. Find the point
A survey is taken to estimate the mean annual family income for families living in public housing in Chicago. For a random sample of 29 families, the annual incomes (in hundreds of dollars) are as follows: 90 77 100 83 64 78 92 73 122 96 60 85 86 108 70 139 56 94 84 111 93 120 70 92 100 124 59 112
The GSS asked in 2008, €œOn the average day about how many hours do you personally watch television?€ Software reports the results for females,a. Would you expect that TV watching has a normal distribution? Why or why not? b. On what assumptions is the confidence interval shown
In response to the statement on a recent General Social Survey, €œA preschool child is likely to suffer if his or her mother works,€ suppose the response categories (strongly agree, agree, disagree, strongly disagree) had counts (104, 370, 665, 169). Scores (2, 1,-1,-2) were
The 2008 GSS asked, “What is the highest grade that you finished and got credit for?” (variable EDUC). Of 2018 respondents, the mean was 13.4, the standard deviation was 3.1, and the proportion who gave responses below 12 (i.e., less than a high school education) was 0.166. Explain how you
Refer to the previous exercise. For the 280 African Americans in the 2008 GSS, a 99% confidence interval for the mean of EDUC is (12.38, 13.16). Explain why the following interpretation is incorrect: 99% of all African Americans have completed grades between 12.38 and 13.16.
The 2008 General Social Survey asked respondents how many sex partners they had in the previous 12 months (variable PARTNERS). Software summarizes the results of the responses bya. Based on the reported sample size and standard deviation, verify the reported value for the standard error. b. Based
In December 2004, a report based on the National Survey on Drug Use and Health estimated that 20% of all Americans of ages 16 to 20 drove under the influence of drugs or alcohol in the previous year (AP, December 30, 2004). A public health unit in Wellington, New Zealand, plans a similar survey for
The June 2003 report on Views of a Changing World , conducted by the Pew Global Attitudes Project ( www.people-press.org ), discussed changes in views of the United States by other countries. In the largest Muslim nation, Indonesia, a poll conducted in May 2003 after the Iraq war began reported
A tax assessor wants to estimate the mean property tax bill for all homeowners in Madison, Wisconsin. A survey 10 years ago got a sample mean and standard deviation of $1400 and $1000.a. How many tax records should the tax assessor randomly sample for a 95% confidence interval for the mean to have
A football coach decides to estimate the kicking accuracy of a player who wants to join the team. Of 10 extra point attempts, the player makes all 10. a. Find an appropriate 95% confidence interval for the probability that the player makes any given extra point attempt. b. What’s the lowest value
For parts a and b, is the statement a null hypothesis, or an alternative hypothesis? a. In Canada, the proportion of adults who favor legalized gambling equals 0.50. b. The proportion of all Canadian college students who are regular smokers is less than 0.24, the value it was 10 years ago. c.
You plan to apply significance testing to your own experiment for testing astrology, in which astrologers have to guess which of four personality profiles is the correct one for someone who has a particular horoscope. Define notation and state hypotheses, letting one hypothesis reflect the
An article in a marketing journal states that “no statistically significant difference was found between men and women in the proportion who said they would consider buying a hybrid car (P-value = 0.63).” In practical terms, how would you explain to someone who has not studied statistics what
A study in Finland in 1998 suggested that vitamin E pills reduced the risk of prostate cancer in a group of male smokers. This was only one effect in a series of reported successes for vitamin E and other health conditions. A more recent study reported that selenium or vitamin E, used alone or in
When medical stories in the mass media report dangers of certain agents (e.g., coffee drinking), later research often suggests that the effects may not exist or are weaker than first believed. Explain how this could happen if some journals tend to publish only statistically significant results.
An alternative hypothesis states that a newly developed drug is better than the one currently used to treat a serious illness. If we reject H0, the new drug will be prescribed instead of the current one. a. Why might we prefer to use a smaller significance level than 0.05, such as 0.01? b. What is
Explain why the terminology “do not reject H0” is preferable to “accept H0.”
It is more informative and potentially less misleading if you conclude a test by reporting and interpreting the P-value rather than by merely indicating whether or not you reject H0 at the 0.05 significance level. One reason is that a reader can then tell whether the result is significant at any
Explain the difference between statistical significance and practical significance. Make up an example to illustrate your reasoning.
An advertisement by Company A says that three of every four doctors recommend pain reliever A over all other brands combined. a. If the company based this claim on interviewing a random sample of doctors, explain how they could use a significance test to back up the claim. b. Explain why this claim
Consider the medical diagnosis of breast cancer with mammograms. An AP story (September 19, 2002) said that a woman has about a 50% chance of having a false-positive diagnosis over the course of 10 annual mammography tests. Relate this result to the chance of eventually making a Type I error if you
For a test of H0: p = 0.50, the z test statistic equals 1.04. a. Find the P-value for Ha: p > 0.50. b. Find the P-value for Ha: p ≠ 0.50. c. Find the P-value for Ha: p < 0.50. d. Do any of the P-values in part a, part b, or part c give strong evidence against H0? Explain.
A random sample of size 1000 has x = 104. The significance level α is set at 0.05. The P-value for testing H0: μ = 100 against Ha: μ ≠ 100 is 0.057. Explain what is incorrect about each of the following interpretations of this P-value, and provide a proper interpretation. a. The probability
One interpretation for the P-value is that it is the smallest value for the significance level α for which we can reject H0. Illustrate using the P-value of 0.057 from the previous exercise.
Refer to the previous exercise. Suppose z = 2.50 instead of 1.04. a. Find the P-value for (i) Ha: p > 0.50. (ii) Ha: p ≠ 0.50, and (iii) Ha: p < 0.50. b. Do any of the P-values in part a provide strong evidence against H0? Explain.
Suppose you wanted to test H0: p = 0.50, but you had 0 successes in n trials. If you had found the test statistic using the se = √p̂(1 - p̂)/n designed for confidence intervals, show what happens to the test statistic. Explain why se0 = √p0(1 - p0)/n is a more appropriate se for tests.
A medical researcher conducts a significance test whenever she analyzes a new data set. Over time, she conducts 100 independent tests. a. Suppose the null hypothesis is true in every case. What is the distribution of the number of times she rejects the null hypothesis at the 0.05 level? b. Suppose
For a test of H0: p = 0.50, the sample proportion is 0.35 based on a sample size of 100. a. Show that the test statistic is z = -3.0. b. Find the P-value for Ha: p 6 0.50. c. Does the P-value in part b give much evidence against H0? Explain.
A recent study 6 considered whether dogs could be trained to detect if a person has lung cancer or breast cancer by smelling the subject’s breath. The researchers trained five ordinary household dogs to distinguish, by scent alone, exhaled breath samples of 55 lung and 31 breast cancer patients
Americans ages 18 to 29 are considered to be less religious than older Americans. According to recent studies by the Pew Forum on Religion & Public Life, fewer young adults are affiliated with a specific religion than older people today. And, compared with their elders, fewer young people say that
Examples 1, 3, and 5 referred to a study about astrology. Another part of the study used the following experiment: Professional astrologers prepared horoscopes for 83 adults. Each adult was shown three horoscopes, one of which was the one an astrologer prepared for them and the other two were
Examples 2 and 6 described a study about therapeutic touch (TT). A second run of the same experiment in the study used 13 TT practitioners who had to predict the correct hand in each of 10 trials. a. Defining notation, set up hypotheses to test that the probability of a correct guess is 0.50
Studies that compare treatments for chronic medical conditions such as headaches can use the same subjects for each treatment. This type of study is commonly referred to as a crossover design. With a crossover design, each person crosses over from using one treatment to another during the study.
For a large supermarket chain in Florida, a women€™s group claimed that female employees were passed over for management training in favor of their male colleagues. The company denied this claim, saying they picked the employees from the eligible pool at random to receive this training.
Refer to the previous exercise. a. Explain why the alternative hypothesis of bias against males is Ha: p 6 0.60. b. Show that the P-value for testing H0: p = 0.60 against Ha: p 6 0.60 equals 0.90. Interpret. Why is it large?
A study (J. Amer. Med. Assoc., vol. 284, p. 831, 2000) considered whether daily consumption of 1200 mg of garlic could reduce tick bites. The study used a crossover design with a sample of Swedish military conscripts, half of whom used placebo first and garlic second and half the reverse. The
According to an exit poll in the 2008 Vermont gubernatorial election, 54.5% of the sample size of 837 reported voting for the Republican candidate Douglas. Is this enough evidence to predict who won? Test that the population proportion who voted for Douglas was 0.50 against the alternative that it
The 49 students in a class at the University of Florida made blinded evaluations of pairs of cola drinks. For the 49 comparisons of Coke and Pepsi, Coke was preferred 29 times. In the population that this sample represents, is this strong evidence that a majority prefers one of the drinks? Refer to
A fast-food chain wants to compare two ways of promoting a new burger (a turkey burger). One way uses a coupon available in the store. The other way uses a poster display outside the store. Before the promotion, their marketing research group matches 50 pairs of stores. Each pair has two stores
Example 4, on whether dogs can detect bladder cancer by selecting the correct urine specimen (out of seven), used the normal sampling distribution to find the P-value. The normal distribution P-value approximates a P-value using the binomial distribution. That binomial P-value is more appropriate
Refer to the previous exercise. If the same sample mean and standard deviation had been based on n = 5 instead of n = 20, the test statistic would have been t = 1.20. a. Would the P-value for Ha: μ ≠ 100 be larger, or smaller, than when t = 2.40? Why? b. Other things being equal, explain why
In a recent study, 10 272 moderately obese subjects were randomly assigned to one of three diets: low-fat, restricted-calorie; Mediterranean, restricted calorie; or low-carbohydrate, non-restricted- calorie. The prediction was that subjects on a low- carbohydrate diet would lose weight, on the
When the 636 male workers in the 2008 GSS were asked how many hours they worked in the previous week, the mean was 45.5 with a standard deviation of 15.16. Does this suggest that the population mean work week for men exceeds 40 hours? Answer by: a. Identifying the relevant variable and
When the 127 workers aged 18–25 in the 2008 GSS were asked how many hours they worked in the previous week, the mean was 37.47 with a standard deviation of 13.63. Does this suggest that the population mean work week for this age group differs from 40 hours? Answer by: a. Identifying the relevant
An industrial plant claims to discharge no more than 1000 gallons of wastewater per hour, on the average, into a neighboring lake. An environmental action group decides to monitor the plant, in case this limit is being exceeded. Doing so is expensive, and only a small sample is possible. A random
A disadvantage of the experimental design in Example 8 on weight change in anorexic girls is that girls could change weight merely from participating in a study. In fact, girls were randomly assigned to receive a therapy or to serve in a control group, so it was possible to compare weight change
A crossover study of 13 children suffering from asthma (Clinical and Experimental Allergy, vol. 20, pp. 429€“432, 1990) compared single inhaled doses of formoterol (F) and salbutamol (S). The outcome measured was the child€™s peak expiratory flow (PEF) 8 hours following
Wine-pouring vending machines, previously available in Europe and international airports, have become popular in the last few years in the United States. They are even approved to dispense wine in some Wal-Mart stores. The available pouring options are a 5-ounce glass, a 2.5-ounce half-glass, and a
In Exercise 9.24, a fast-food chain compared two ways of promoting a turkey burger. In a separate experiment with 10 pairs of stores, the difference in the month€™s increased sales between the store that used coupons and the store with the outside poster had a mean of $3000. Does this
Refer to the previous exercise. a. Explain how the result of the 95% confidence interval shown in the table agrees with the test decision using the 0.05 significance level. b. Suppose you instead wanted to perform a one-sided test, because the study predicted that the increase in sales would be
Example 8 described a study about various therapies for teenage girls suffering from anorexia. For each of 17 girls who received the family therapy, the changes in weight were 11, 11, 6, 9, 14, -3, 0, 7, 22, -5, -4, 13, 13, 9, 4, 6, 11. a. Plot these data with a dot plot or box plot, and
Suppose the mean GPA of all students graduating from the University of Alabama in 1985 was 3.05. The registrar plans to look at records of students graduating in 2011 to see if mean GPA has changed. Define notation and state the null and alternative hypotheses for this investigation.
Ideally, results of a statistical analysis should not depend greatly on a single observation. To check this, it€™s a good idea to conduct a sensitivity study. This entails redoing the analysis after deleting an outlier from the data set or changing its value to a more typical value and
Results of 99% confidence intervals are consistent with results of two-sided tests with which significance level? Explain the connection.
In the experiment in Example 4, we got a P-value = 0.000 for testing H0: p = 1/7 about dogs’ ability to diagnose urine from bladder cancer patients. a. For the significance level 0.05, what decision would you make? b. If you made an error in part a, what type of error was it? Explain what the
Example 3, in testing H0: p = 1/3 against Ha: p 7 1/3, analyzed whether astrologers could predict the correct personality chart (out of three possible ones) for a given horoscope better than by random guessing. In the words of that example, what would be(a) A Type I error and(b) A Type II error?
Example 8 tested a therapy for anorexia, using hypotheses H0: μ = 0 and Ha: μ ≠ 0 about the population mean weight change μ. In the words of that example, what would be(a) A Type I error and(b) A Type II error?
Refer to the previous exercise. When we test H0: μ = 0 against Ha: μ > 0, we get a P-value of 0.02. a. What would the decision be for a significance level of 0.05? Interpret in context. b. If the decision in part a is in error, what type of error is it? c. Suppose the significance level were
Consider the test of H0: The defendant is not guilty against Ha: The defendant is guilty. a. Explain in context the conclusion of the test if H0 is rejected. b. Describe the consequence of a Type I error. c. Explain in context the conclusion of the test if you fail to reject H0. d. Describe the
Consider the test of H0: The new drug is safe against Ha: the new drug is not safe. a. Explain in context the conclusion of the test if H0 is rejected. b. Describe the consequence of a Type I error. c. Explain in context the conclusion of the test if you fail to reject H0. d. Describe the
Consider medical diagnostic testing, such as using a mammogram to detect if a woman may have breast cancer. Define the null hypothesis of no effect as the patient does not have the disease. Define rejecting H0 as concluding that the patient has the disease. See the table for a summary of the
A study plans to have a sample of obese adults follow a proposed low-carbohydrate diet for three months. The diet imposes limited eating of starches (such as bread and pasta) and sweets, but otherwise no limit on calorie intake. Consider the hypothesis, The population mean of the values of weight
Refer to the previous exercise about medical diagnoses. A New York Times article (February 17, 1999) about the PSA blood test for detecting prostate cancer stated: “The test fails to detect prostate cancer in 1 in 4 men who have the disease.” a. For the PSA test, explain what a Type I error is,
Which error, Type I or Type II, would usually be considered more serious for decisions in the following tests? Explain why. a. A trial to test a murder defendant’s claimed innocence, when conviction results in the death penalty. b. A medical diagnostic procedure, such as a mammogram.
Two researchers conduct separate studies to test H0: p = 0.50 against Ha: p ≠ 0.50, each with n = 400. a. Researcher A gets 220 observations in the category of interest, and p̂ = 220/400 = 0.550 and test statistic z = 2.00. Show that the P-value = 0.046 for Researcher A’s analysis. b.
A study considers if the mean score on a college entrance exam for students in 2010 is any different from the mean score of 500 for students who took the same exam in 1985. Let μ represent the mean score for all students who took the exam in 2010. For a random sample of 25,000 students who took
Example 11 analyzed political conservatism and liberalism in the United States. Suppose that the sample mean of 4.11 and sample standard deviation of 1.43 were from a sample size of only 25, rather than 1933. a. Find the test statistic. b. Find the P-value for testing H0: μ = 4.0 against Ha: μ
In 2004, New York Attorney General Eliot Spitzer filed a lawsuit against GlaxoSmithKline pharmaceutical company, claiming that the company failed to publish results of one of their studies that showed that an antidepressant drug (Paxil) may make adolescents more likely to commit suicide. Partly as
Refer to Example 12. Using a tree diagram, given that H0 is rejected, approximate P(Type I error) under the assumption that a true effect exists 20% of the time and that there’s a 30% chance of a Type II error.
a. An advertisement by Schering Corp. in 1999 for the allergy drug Claritin mentioned that in a clinical trial, the proportion who showed symptoms of nervousness was not significantly greater for patients taking Claritin than for patients taking placebo. Does this mean that the population
A study is designed to test H0: p = 0.50 against Ha: p > 0.50, taking a random sample of size n = 100, using significance level 0.05. a. Show that the rejection region consists of values of p̂ 7 0.582. b. Sketch a single picture that shows (i) the sampling distribution of p̂ when H0 is true and
Exercise 9.19 tested the claim that female employees were passed over for management training in favor of their male colleagues. Statewide, the large pool of more than 1000 eligible employees who can be tapped for management training of selecting a female for any given selection. For testing H0: p
Recall that for the same sample size the smaller the probability of a Type I error, α, the larger the P(Type II error). Let’s check this for Example 13. There we found P(Type II error) for testing H0: p = 1>3 (astrologers randomly guessing) against Ha: p > 1 > 3 when actually p = 0.50, with n =
For testing H0: p = 1 / 3 (astrologers randomly guessing) against Ha: p > 1 / 3 with n = 116, Example 13 showed that P(Type II error) = 0.02 when p = 0.50. Now suppose that p = 0.35. Recall that P(Type I error) = 0.05. a. Show that P(Type II error) = 0.89. b. Explain intuitively why P(Type II
In Example 13 for testing H0: p = 1/3 (astrologers randomly guessing) with n = 116 when actually p = 0.50, suppose we used Ha: p ≠ 1/3. Then show that: a. A Type II error occurs if 0.248 < p̂ < 0.419. b. The probability is 0.00 that p̂ < 0.248 and 0.96 that p̂ 7 0.419. c. P(Type II error) =
Consider Example 14 about the power of the test used in the TT experiment for testing H0: p = 0.50 against Ha: p > 0.50, where p is the probability of a correct prediction about which hand was nearer the researcher’s hand. In a significance test planned for a second set of trials, the power was
Refer to the simulation in Activity 2 at the end of the section. a. Repeat the simulation, now assuming that actually p = 0.45. In a large number of simulations, what proportion of the time did you make a Type II error at the significance level of 0.05? What does theory predict for this
For each of the following hypotheses, explain whether it is a null hypothesis or an alternative hypothesis: a. For females, the population mean on the political ideology scale is equal to 4.0. b. For males, the population proportion who support the death penalty is larger than 0.50. c. The diet has
A person who claims to possess extrasensory perception (ESP) says she can guess more often than not the outcome of a flip of a balanced coin. Out of 20 flips, she guesses correctly 12 times. Would you conclude that she truly has ESP? Answer by reporting all five steps of a significance test of the
Consider all cases in which a pro basketball player shoots two free throws and makes one and misses one. Which do you think is more common: making the first and missing the second, or missing the first and making the second? One of the best shooters was Larry Bird of the Boston Celtics. During
California’s governor election in 2010 had two major candidates, Brown and Whitman. a. For a random sample of 650 voters in one exit poll, 360 voted for Brown and 240 for Whitman. Conduct all five steps of a test of H0: p = 0.50 against Ha: p ≠ 0.50, where p denotes the probability that a
When the 2000 General Social Survey asked, €œWould you be willing to pay much higher taxes in order to protect the environment?€ (Variable GRNTAXES), 369 people answered yes and 483 answered no. (We exclude those who made other responses.) Let p denote the population proportion
A Pew Research Center poll in 2010 asked a random sample of 2505 adults about their attitudes and opinions concerning the U.S. government (www.people-press.org). When asked whether they felt content, frustrated or angry, 56% said that they were frustrated. Let p denote the population proportion who
In an experiment on chlorophyll inheritance in maize (corn), of the 1103 seedlings of self fertilized green plants, 854 seedlings were green and 249 were yellow. Theory predicts the ratio of green to yellow is 3 to 1. Show all five steps of a test of the hypothesis that 3 to 1 is the true ratio.
You are told that a ball will be randomly drawn from one of two boxes (A and B), both of which contain black balls and red balls, and if a red ball is chosen, you will win $100. You are also told that Box A contains half black balls and half red balls, but you are not told the proportions in Box B.
A fraternity at a university lobbies the administration to start a hockey team. To bolster its case, it reports that of a simple random sample of 100 students, 83% support starting the team. Upon further investigation, their sample has 80 males and 20 females. Should you be skeptical of whether the
A bank wants to evaluate which credit card would be more attractive to its customers: One with a high interest rate for unpaid balances but no annual cost, or one with a low interest rate for unpaid balances but an annual cost of $40. For a random sample of 100 of its 52,000 customers, 40 say they
A jury list contains the names of all individuals who may be called for jury duty. The proportion of the available jurors on the list who are women is 0.53. If 40 people are selected to serve as candidates for being picked on the jury, show all steps of a significance test of the hypothesis that
Refer to the previous exercise. a. Explain what Type I and Type II errors mean in the context of that exercise. b. If you made an error with the decision in part d, is it a Type I or a Type II error?

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