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nonparametric statistical inference

Statistical Methods For The Social Sciences 4th Edition Barbara Finlay, Alan Agresti - Solutions

7.6. The U.S. Census Bureau reported that in 2002 the median net worth in the U.S. was estimated to be about $89,000 for white households and $6000 for black households.(a) Identify the response variable and the explanatory variable.(b) Compare the groups using a (i) difference,(ii) ratio.
7.5. The National Center for Health Statistics recently estimated thatthe mean weight for adult American women was 140 pounds in 1962 and 164 pounds in 2002.(a) Suppose these estimates had standard errors of 2 pounds each year. Estimate the increase in mean weight in the population from 1962 to
7.4. When a recent Eurobarometer survey asked subjectsin each European Union country whetherthey would be willing to pay more for energy produced from renewable sources than for energy produced from other sources, the proportion answering yes varied from a high of 0.52 in Denmark (« = 1008)to a
7.3. The National Health Interview Survey (www.cdc.gov/nchs) estimated that current cigarette smokers were 41.9% of American adults in 1965 and 21.5%in 2003.(a) Estimate the difference between the proportions who smoked in the two years.(b) Suppose the standard error were reported as 0.020 for each
7.2. Transatlantic Trends is an annual survey of American and European public opinion (see www.transatlantictrends.org), with a random sample of about 1000 adults from each of 13 European countries each year. In 2002, 38% of Europeans expressed a positive attitude about President George W. Bush's
7.1. An Associated Press story (Feb. 23, 2007) about UCLA's annual survey of college freshmen indicated that 73% of college freshmen in 2006 considered being financially well offto be very important, compared to 42% in 1966 (the first year the survey was done). It also reported that 81% of 18- to
6.67. Refer to the ESPexperiment in Exercise 6.33, with n = 5.(a) For what value(s) of jc = number of correct guesses can you reject Hq: tt = 0.50 in favor of Ha: tt > 0.50, using a = 0.05?(b) For what valuc(s) ofx can you reject Hq using a = 0.01? (Ao/e; Forsmallsamples, it may not be possible to
6.66. You test//q: tt = 0.50 against//rt: tt > 0.50, using a = 0.05. In fact, H„ is true. Explain why /'(Type 11 error) increases toward 0.95 as tt moves down toward 0.50. (Assume n and a stay fixed.)
6.65. Suppose you wanted to test Hq: tt = 0.50, but of the « = 30 observations, 0 were in the category of interest. If you found the z test statistic using the se = l — tt)//? for confidence intervals, show what happens to the test statistic. Explain why the,=(1 — 7ro)/n is more appropriate
6.64. Each year in Liverpool, New York, a public librarian estimates the mean number of times the books in that library have been checked out in the previous year. To do this, the librarian randomly samples computer records for 100 books and forms a 95%confidence interval for the mean. This has
6.62. Refer to the previous exercise and the P-value of 0.057.(a) Explain why the P-value is the smallest alevel at which Hq can be rejected; that is. P equals the smallest level at which the data are significant.(b) Refer to the correspondence between results of confidence intervals and two-sided
6.61. A random sample of size 40 has y = 120. The Pvalue for testing Hq: p. = 100 against/7„: p, * 100 is P = 0.057. Explain what is incorrect about each of the following interpretations of this P-value, and provide a proper interpretation.(a) The probability that the null hypothesis is correct
6.60. Your friend plans to survey students in your college to study whether a majority feel that the legal age for drinking alcohol should be reduced. He has never studied statistics. How would you explain to him the concepts of(a) null and alternative hypotheses,(b) P-value,(c) a-lcvcl,(d) Type 11
6.59. Explain why the terminology "do not reject F/q" is preferable to "accept Hq."
6.58. Explain the difference between one-sided and twosided alternative hypotheses, and explain how this affects calculation of the P-value.
6.57. Answer true or false for each of the following, and explain your answer:(a) P(Type 11 error) = 1 - P(Type 1 error).(b) If we reject Hq using a = 0.01, then we also reject it using a = 0.05.(c) The P-value is the probability that Hq is true.(Hint: Do wc find probabilities about variables and
6.56. Let (5 denote P(Typc II error). For an a = 0.05-level test of Hq: p, = 0 against Ha: p. >f) with« = 30 observations. /3 = 0.36 at/x = 4. Then(a) At/x = 5, /3 > 0.36.(b) If a = 0.01, then at /x = 4, /3 > 0.36.(c) If n = 50, then at /x = 4, /3 > 0.36.(d) The power of the test is 0.64 at /x =
6.55. A 95% confidence interval for/x is (96,110). Which two statements about significance testsforthe same data are correct?(a) In testing Hq\ p. = 100 against Ha', p. 100, P > 0.05.(b) In testing Hq: /x = 100 against Ha: p # 100, P < 0.05.(c) In testing Hq: p. = /xq against //„: p. /xq, P >
6.54. In the previous exercise, suppose the test statistic t = 3.29.(a) There is strong evidence that /r = 100.(b) There is strong evidence that p. > 100.(c) There is strong evidence that p. < 100.
6.53. The F-value for testing Hq: p. = 100 against Ha:p ¥= 100 is P = 0.001. This indicates that(a) There is strong evidence that p. = 100.(b) There is strong evidence that p, * 100.(c) There is strong evidence that p. > 100.(d) There is strong evidence that p. < 100.(e) If p. were equal to 100,
6.52. We analyze whether the true mean discharge of wastcwatcr per hour from an industrial plant exceeds the company claim of 1000 gallons. For the decision in the one-sided test using a = 0.05:(a) Ifthe plantis not exceeding the limit, but actually p - 1000, there is only a 5% chance that we will
6.51. Some journals have a policy of publishing research results only if they achieve statistical significance at the 0.05 a-level.(a) Explain the dangers of this.(b) When medicalstoriesin the mass media report supposed large dangers or benefits of certain agents (e.g.. coffee drinking, fiber in
6.50. A research study conducts 60 significance tests. Of these, 3 are significant atthe 0.05 level. The authors write a report stressing only the three "significant"results, not mentioning the other 57 tests that were"notsignificant." Explain whatis misleading about their report.
6.49. You conduct a significance test using software. The output reports a F-value of 0.4173545. In summarizing your analyses in a research article, explain why it makes more sense to report P = 0.42 rather than P = 0.4173545.
6.48. An article in a political science journal states that"no significant difference was found between men and women in their voting rates (P = 0.63)." Can wc conclude that the population voting rates arc identical for men and women? Explain.
6.47. An article in a sociology journal that deals with changes in religious beliefs over time states,"For these subjects, the difference in their mean responses on the scale ofreligiosity between age 16 and the current survey wassignificant {P < 0.05)."(a) Explain what it means for the result to
6.46. Medical tests for diagnosing conditions such as breast cancer arc fallible, just like decisions in significance tests. Identify (Hq true, Hq false) with disease (absent, present), and (Reject Ho, Do not reject Hq) with diagnostic test (positive, negative), where a positive diagnosis means
6.45. Consider the analogy between making a decision in a test and making a decision aboutthe innocence or guilt of a defendant in a criminal trial.(a) Explain what Type 1 and Type II errors are in the trial.(b) Explain intuitively why decreasing P(Type I error) increases /'(Type II error).(c)
6.44. In making a decision in a test, a researcher worries aboutthe possibility ofrejecting Hq when it is actually true. Explain how to control the probability of this type of error.
6.43. 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:Redo the analysis after deleting an outlier from the data set or changing its value to a more typical value, and check whether results
6.42. An experiment with 26 students in an Israeli classroom consisted of giving everyone a lottery ticket, and then later asking if they would be willing to exchange their ticket for another one, plus a small monetary incentive. Only 7 students agreed to the exchange. In a separate experiment, 31
6.41. In the 1990s, the U.S. Justice Department and other groups studied possible abuse by Philadelphia police officers in their treatment of minorities.One study, conducted by the American Civil Liberties Union, analyzed whether African-American drivers were more likely than others in the
6.40. 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?
6.39. A study considered the effects of a special class designed to improve children's verbal skills. Each child took a verbal skills test before and after attendingtheclassforthree weeks. Lety = second exam score — first exam score. The scores on y for a random sample of four children having
6.38. Refer to the data file your class created in Exercise 1.12 (page 9). For variables chosen by your instructor, state a research question and conduct inferential statistical analyses. Also use graphical and numerical methods presented earlier in this text to describe the data and, if necessary,
6.37. Refer to the "Student survey" data file (Exercise 1.11 on page 8).(a) Test whether the population mean political ideology differs from 4.0. Report the P-valuc, and interpret.(b) Test whether the proportion favoring legalized abortion equals, or differs from, 0.50.Report the P-valuc, and
6.36. You can use an applet to repeatedly generate random samples and conduct significance tests, to illustrate their behavior when used for many samples. To try this, go to the significance test for a proportion applet at www.prenhall.com/???. Set the null hypothesis as Hq: tt = 1/3 for a
6.35. In a given year, the probability that an American female dies in a motor vehicle accident equals 0.0001 {Statistical Abstract ofthe United States).(a) In a city having 1 million females, find the mean and standard deviation of x = number of deaths from motor vehicle accidents. State the
6.34. In a CNN exit poll of 1336 voters in the 2006 Senatorial election in New York State, let jc = number in exit poll who voted for the Democratic candidate, Hillary Clinton.(a) Explain why this scenario would seem to satisfy the three conditions needed to use the binomial distribution.(b) If the
6.33. A person claiming to possess extrasensory perception (ESP) says she can guess more often than not the outcome of a flip of a balanced coin in another room, not visible to her.
6.32. A jury list contains the names of all individuals who may be called forjury duty. The proportion of women on the list is 0.53. Ajury ofsize 12 isselected at random from the list. None selected arc women.(a) Find the probability ofselecting 0 women.(b) Test the hypothesis that the selections
6.31. Refer to the previous exercise.(a) Find P(Type II error) if /x — 5. How does P(Typc II error) depend on the value of fx"?(b) Find P(Typc II error) if/x = 10 and a = 0.01.How does P(Type II error) depend on a?(c) How does P(Type 11 error) depend on 77?
6.30. Studies have considered whether neonatal sex differences exist in behavioral and physiological reactionsto stress. One study6 evaluated changesin heart rate for a sample ofinfants placed in a stressfulsituation. The sample mean change in heart rate was small for males compared to females:
6.29. Let 77 denote the proportion ofschizophrenics who respond positively to treatment. A testis conducted of Hq: tt = 0.50 against Ha: v > 0.50, for a sample ofsize 25, using a = 0.05.(a) Find the region of sample proportion values for which Ho is rejected.(b) Suppose thattt = 0.60. Find /"(Type
6.28. A decision is planned in a test of Hq: /x = 0 against Ha: fx > 0, using a = 0.05. If /r = 5, P(Type II error) = 0.17.(a) Explain the meaning of this last sentence.(b) If the test used a = 0.01, would P(Typc II error) be less than, equal to, or greater than 0.17? Explain.(c) Ifix = 10, would
6.27. Refer to Example 6.8 on "medical discoveries"(page 165). Using a tree diagram, approximate P(Type I error) under the assumption that a true effect exists 20% of the time and that P(Type II error) = 0.30.
6.26. A report released on September 25, 2006 by the Collaborative on AcademicCareersin HigherEducation indicated thatthere is a notable gap between female and male academics in their confidence that tenure rules arc clear, with men feeling more confident. The 4500 faculty members in the survey
6.25. A study considers whether the mean score jx on a college entrance exam for students in 2007 is any different from the mean of 500 for students in 1957. Test Ho: fx - 500 against H,,: 500, iffor a nationwide random sample of 10,000 students who took the exam in 2007, y = 497 and s = 100. Show
6..24. Jones and Smith separately conduct studies to test Ho: it = 0.50 against Ho'- tt # 0.50, each with n = 400. Jones gets tt = 220/400 = 0.550. Smith gets tt = 219/400 = 0.5475.(a) Show that z = 2.00 and P-value = 0.046 for Jones. Show that z — 1.90 and P-value =0.057 for Smith.(b) Using a =
6.23. Jones and Smith separately conduct studies to test Hq: p, = 500 against Hq: p # 500, each with n = 1000. Jones gets y = 519.5, with se = 10.0.Smith gets y = 519.7, with se = 10.0.(a) Show that t = 1.95 and P-value = 0.051 for Jones. Show that t = 1.97 and P-value =0.049 for Smith.
6.22. Example 6.4 (page 151) tested a therapy for anorexia, using Hq'. ^ = 0 and Ha'. p, > 0 about the population mean weight change.(a) In the words of that example, what would be a (i) Type I error, (ii) Type II error?(b) The P-value was 0.017. If the decision for a =0.05 were in error, what type
6.21. A multiple-choice test question has four possible responses. The question is difficult, with none of the four responses being obviously wrong, yet with only one correct answer. It first occurs on an exam taken by 400 students. Test whether more people answer the question correctly than would
6.20. The authorship of an old document is in doubt.A historian hypothesizes that the author was a journalist named Jacalyn Levine. Upon a thorough investigation of Levine's known works, it is observed that one unusual feature of her writing was that she consistently began 6% of her sentences with
6.19. A mayoral election in Madison, Wisconsin, has two candidates. Exactly halfthe residents currently prefer each candidate.(a) For a random sample of 400 voters, 230 voted for a particular candidate. Are you willing to predict the winner? Why?(b) For a random sample of40 voters, 23 voted for a
6.18. The previous exercise analyzed whether astrologers could predictthe correct personality chart for a given horoscope better than by random guessing.In the words ofthat study, what would be a(a) Type I error,(b) Type II error?
6.17. In the scientific test of astrology discussed in Example 6.9 (page 166), the astrologers were correct with 40 of their 116 predictions. Test Hq'. tt = 1/3 against He,: tt > 1/3. Find the P-value, make a decision using a = 0.05, and interpret.
6.16. A Pew Research Center poll (May 14, 2003) of 1201 adults asked, "All in all, do you think affirmative action programs designed to increase the number of black and minority students on college campuses arc a good thing or a bad thing?" Sixty percent said good, 30% said bad, and 10% said don't
6.15. When a recent GSSasked, "Would you be willing to pay much higher taxes in order to protect the environment?" (variable GRNTAXES), 369 people answered yes and 483 answered no. Software shows the following results to analyze whether a majority or minority of Americans would answer yes:
6.14. Same-sex marriage was legalized across Canada by the Civil Marriage Act enacted in 2005. Is this supported by a majority, or a minority, ofthe Canadian population? A poll conducted for the Globe and Mail newspaperin July 2005 of 1000 Canadians asked whetherthis bill should stand or be
6.13. For a test of Hq: tt = 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 and interpret the P-value for Ha:TT < 0.50.(c) For a significance level of a = 0.05, what decision do you make?(d) If the decision in (c) was in
6.12. For a test of Hq: tt = 0.50, the z test statistic equals 1.04.(a) Find the P-value for Ha: tt > 0.50.(b) Find the P-value for H,,: tt # 0.50.(c) Find the P-value for Ha: tt < 0.50.(d) Do any of the P-values in (a), (b), or (c) give strong evidence against Hq7 Explain.
6.11. Results of 99% confidence intervals for means are consistent with results oftwo-sided tests with which a-level? Explain the connection.
6.10. In Example 6.2 on political ideology (page 149), suppose we use the scores (-3, —2, -1, 0, 1, 2, 3) instead of the scores (1, 2, 3, 4, 5, 6, 7) used there. We then test Hq: /x - 0. Explain the effect of the change in scores on (a) the sample mean and standard deviation, (b) the test
6.9. In response to the statement "A preschool child is likely to suffer if his or her mother works," the response categories (Strongly agree. Agree. Disagree, Strongly disagree) had counts (91, 385, 421, 99) for responses in a GSS. To treat this ordinal variable as quantitative, we assign scoresto
6.8. By law, an industrial plant can discharge no more than 500 gallons of waste water per hour, on the average, into a neighboring lake. Based on other infractions they have noticed, an environmental action group believes this limit is being exceeded.Monitoring the plant is expensive, and a random
6.7. 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 /x for female employees matches this norm. For a random sample of nine female
6.6. Example 6.4 (page 151) described a study about therapies for teenage girls suffering from anorexia.For the 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.Part of an SPSS printout for the data shows:
6.5. Find and interpret the P-valuc for testing Hq:/j. = 100 against Hq : /i =£ 100, if a sample has(a) n = 400, y = 103,5 = 40.(b) n = 1600, y = 103, 5 = 40. Comment on the effect of n on the results of a significance test.
6.3. For a test of Hq: /x - 0 against Ha: /x ^ 0 with n = 1000, the I test statistic equals 1.04.(a) Find the P-valuc and interpret it. (Note: You can use the standard normal to approximate the t distribution.)(b) Suppose t = -2.50 rather than 1.04. Find the P-value. Does this provide stronger, or
6.2. You want to know whether adults in your country think the ideal number of children is equal to 2, or higher or lower than that.(a) Define notation and state the null and alternative hypotheses for studying this.(b) For responses in a recent GSS to the question,"What do you think is the ideal
6.1. For (a)-(c), is it 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 now is less than 0.24 (the value it was ten years ago).(c) The mean 10
5.77. To construct a large-sample confidence interval for a proportion tt, it is not necessary to substitute tt for the unknown value of 77 in the formula for the
5.76. To encourage subjects to make responses on sensitive questions, the method ofrandomizedresponse is often used. The subject is asked to flip a coin, in secret. If it is a head, the subject tosses the coin once more and reports the outcome, head or tails.If, instead, the first flip is a tail,
5.75. Let tt be the probability a randomly selected voter prefers the Republican candidate. You sample 2 people, and neither prefers, the Republican. Find the point estimate of tt. Does this estimate seem sensible? Why? (The Bayesian estimatoris an alternative one that uses a subjective approach,
5.74. Find the standard error of the sample proportion when tt — 0 or tt — 1. What does this reflect?
5.72. For a random sample of n subjects, explain why it is about 95% likely that the sample proportion has error no more than 1/ in estimating the population proportion. {Hint: To show this "1/Jli Rule," find two standard errors when tt = 0.50, and explain how this compares to two standard errors
5.71. Refer to the previous exercise. Provide the proper interpretation.
5.68. Other things being equal, quadrupling the sample size causes the width of a confidence interval to(a) double(b) halve(c) be one quarter as wide(d) stay the same.
5.67. Increasing the confidence level causes the width of a confidence interval to(a) increase(b) decrease(c) stay the same.
5.65. To use the large-sample confidence interval for tt, you need at least 15 outcomes of each type. Show that the smallest value of n for which the method can be used is (a) 30 when tt = 0.50, (b) 50 when tt = 0.30, (c) 150 when tt = 0.10. That is, the overall n must increase as tt moves toward 0
5.64. The 2006 publication Attitudes towards European Union Enlargement from Eurobarometer states, The readers are reminded that survey results arc estimations, the accuracy of which rests upon the sample size and upon the observed percentage. With samples of about 1,000 interviews, the real
5.61. How does population heterogeneity affectthe sample size required to estimate a population mean?Illustrate with an example.
5.60. Give an example of a study in which it would be important to have(a) A high degree of confidence(b) A high degree of precision
5.58. Explain why confidence intervals are wider with(a) larger confidence levels,(b) smaller sample sizes.5.59. Why would it be unusual to see a(a) 99.9999%,(b) 25% confidence interval?
5.56. What is the purpose of forming a confidence interval for a parameter? What can you learn from it that you could not learn from a point estimate of the parameter?5.57. An interval estimate for a mean is more informative than a point estimate, because with an interval estimate you can figure
5.54. The observations on TV watching for the seven Muslims in the GSSin a recent yearwere 0,0,2,2,2, 4, 6. A 95% confidence interval for the population
A preschool child is likely to suffer if her mother works: (776,1054)Analyze these data. Prepare a one-page report stating assumptions, showing results of description and inference, and summarizing conclusions.
It is better for everyone involved ifthe man is the achiever outside the home and the woman takes care of the home and the family: (627, 1208)
Women should take care of running their homes and leave running the country up to men: (275,1556)
5.53. When subjectsin a recent GSS were asked whether they agreed with the following statements, the(yei1, no) counts under various conditions were as follows:
5.51. In 2006, the GSS asked about the number of hours a week spent on the World Wide Web. excluding e-mail (variable denoted WWWHR). State a research question you could address about this response variable and a relevant explanatory variable. Go to sda.bcrkclcy.cdu/GSS and analyze the data.
5.48. Refer to the previous exercise. Using the confidence interval for a proportion applet, let's check that the confidence interval for a proportion may work poorly with small samples. Set « = 10 and tt = 0.90. Click on Simulate to generate 100 random samples, each of size 10, forming confidence
5.47. You can use an applet to repeatedly generate random samples and construct confidence intervals, to
5.46. Refer to Example 5.9 (page 130). Construct a 95%confidence interval for the median time since a book was last checked out. Interpret.
5.44. You randomly sample five students at your school to estimate the proportion of students who like tofu. None of the five students say they like it.(a) Find the sample proportion who like it and its standard error. Does the usual interpretation ofse make sense?(b) Why is it not appropriate to
5.36. A television network plans to predict the outcome of an election between Jacalyn Levin and Roberto 5.37. A public health unit wants to sample death records for the past year in Toronto to estimate the proportion of the deaths that were due to accidents.Health officials want the estimate to be
5.33. A study estimates the mean annual family income for families living in public housing in Chicago.For a random sample of 30 families, the annual incomes (in hundreds of dollars) are 83 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 79.(a) Construct
3.50 with standard deviation 1.51?(c) What assumption are you making about the scale of measurement for political ideology when you use the sample mean and standard deviation?
5.31. The GSS 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 2004 GSS gets software output:Variable N Mean StDev SE Mean 997, CI Polviews 1294 4.23 1.39 0.0387
5.30. For the "Florida student survey" data file mentioned in Exercise 1.11, software reportsthe results for responses on the number of times a week the subject reads a newspaper:Variable N Mean Std Dev SE Mean 95.07. CI News 60 4.1 3.0 0.387 (3.32, 4.88)(a) Interpret the confidence interval

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