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statistical sampling to auditing
Statistical Methods For The Social Sciences 3rd Edition Alan Agresti, Barbara Finlay - Solutions
20. Newsweek magazine (March 27. 1989) reported results of a poll about religious beliefs, conducted by the Gallup organization. Of 750 American adults, 24% believed in tein- carnation Treating this as a random sample, construct and interpret a 95% confidence interval for the true proportion of
21. 01 577.006 people involved in motor vehicle accidents in Florida in a recent year. 412,878 were wearing seat belts (Florida Department of Highway Safety and Motor Vehicles).a) Find a point estimate of the population proportion of Florida motorists wearing seal belts.b) Construct a 99%
22. A random sample is taken of students at a large university to determine the proportion. who own automobiles. If 66% of the students in the sample own automobiles. form a 99% confidence interval for the proportion of all students owning automobiles under the following conditionsa) Sample size is
23 Out of an election day sample of 400 individuals who voted in a gubernatorial election. 160 voted for Jones and 240 for Smuth.a) Assuming this is a random sample of all voters, construct a 99% confidence interval for the proportion of votes that Jones will receive Do you think that Jones will
24. A city council votes to appropriate funds for a new civic auditorium. The mayor of the city threatens to veto this decision unless it can be shown that a majority of citizens would use it at least twice a year. The council commissions a poll of city residents. For a random sample of 400
25 Refer to Problem 5.16 Before obtaining the sample, the anthropologist wanted to deter- mine the size of sample needed to estimate to within 07 with 95% confidence. Assum- ing nothing about the value of . calculate the sample size that ensures at least this degree of accuracy
26 To estimate the proportion of traffic deaths in Florida last year that were alcohol related. determine the necessary sample size for the estimate io be accurate to within 06 with probability .90. Based on results of a previous study, we expect the proportion to be about 30
27 A study is being planned for estimating the proportion of marned women with living par- ents in the United States who live in the same state as their parents Approximately how large a sample size is needed to guarantee estimating this proportion correct to withina) .10 with probability 95b) 05
28. A public health unit wants to sample death records for the past year in New York City to estimate the proportion of the deaths that were due to accidents. They want the estimate to be accurate to within 02 with probability 95a) Find the necessary sample size if, based on previous studies they
29. A tax assessor wants to estimate the mean property tax bill for all homeowners living in Madison. Wisconsin. A survey completed three years ago indicated that the mean and standard deviation were $1400 and $1000. Assunning that the standard deviation is still about $1000. how large a sample of
30 An estimate is needed of the mean acreage of farms in Canada. The estimate must be correct to within 25 acres with probability .95. A preliminary study suggests that 200 acres is a reasonable guess lor the standard deviation of farm size. How large a sample of farms is required?
31. Refer to the previous exercise. A sample is selected of the size believed to be needed to estimate the mean correct to within 25 acres with probability .95 Suppose, however, that the sample has a standard deviation of 400 acres, rather than 200. Then, how close can we expect the sample mean to
32 How large a sample size is needed to estimate the mean annual income of Native Ameri- cans correct to within $1000 with probability .99 Suppose there is no prior information about the standard deviation of annual income of Native Americans, but we guess that about 95% of their incomes are
33 A random sample of n = 20 private companies having at least 100 employees each is cho- sen to form the basis of a study of political contributions by industry In particular, a 95% confidence interval is wanted for the median yearly total contribution to political cam- paigns by all companies
34. Refer to Problem 56 Construct a 95% confidence interval for the inedian annual income of the public housing residents. Interpret
35. Refer to Example 5.5. Construct a (a) 90%. (b) 99% confidence interval for the median time since a book was last checked out. Interpret.
36. Refer to the WWW data set (Problem 1.7). Using computci software, construct and in- terpret a 95% confidence interval for the mean weekly number of hours spent watching TV
37. For the WWW data set. find a 95% confidence interval for the proportion believing in life after death interpret.
38 Refe: to the data file created in Problem 1.7 For variables chosen by your instructor, con- duct inferential statistical analyses using basic methods of point and interval estimation. Summarize and interpret your findings
39. Based on data from the National Survey of Families and Households, a recent study (by S South and G Spitze, American Sociological Review, Vol. 59, 1994. pp 327-347) reported the descriptive statistics in Table 5.4 for the hours spent in housework and in employment per week. Analyze these data,
40. The 1994 General Social Survey asked subjects about government spending on the envi- ronment. Table 5.5 shows the responses. Analyze and interpret these data TABLE 5.5 Response Frequency Spending 100 much 130 About right Spending too little 370 880
41. The 1988 General Social Survey asked respondents, "Did you live with your husband/wife before you got manied?" The responses were 176 yes, 566 no. 725 not applicable, and 14 no answer. Analyze these data.
42 Explain what it means for an estimator to be (a) unbiased, (b) efficient.
43. Explain the distinction between point estimation and interval estimation. Why is an in- terval estimate more informative than a point estimate?
44. What is the purpose of forming a confidence interval for a population mean? What does one learn that one could not otherwise learn by calculating a point cstimate, such as the sample mean?
45. Explain why confidence intervals are wide: when we use (a) larger confidence coeffi- cients. (b) smaller sample sizes.
46. Why would it be unusual to sce a 99.9999% confidence interval?
47. Give an example of a study in which it would be important to havea) A high degree of confidenceb) A high degree of precisionc) Both a high degree of confidence and a high degree of precision
48 How does population heterogeneity affect the sample size required to estimate a popula tion nean? Illustrate with an example. Select the best response(s) in Problems 49-51.
49. Increasing the confidence coefficient causes the width of a confidence interval to (a) in- crease, (b) decrease. (c) stay the same
50 Other things being equal, quadrupling the sample size causes the width of a confidence interval to (a) double, (b) halve. (c) be one quartel as wide. (d) stay the same
51. Based on responses of 1467 subjects in General Social Surveys in the mid-1980s, a 95% confidence interval for the mean number of close friends equals (6.8.80). Which of the following interpretations is (are) correct?a) We can be 95% confident that Y is between 68 and 8.0b) We can be 95%
52 A random sample of 50 records yields a 95% confidence interval for the mean age at first mamage of women in a certain county of 21.5 to 23.0 years. Explain what is wrong with each of the following interpretations of this interval.a) If random samples of 50 records were repeatedly selected, then
53. Refer to the previous exercise Provide the proper interpretation.
54. For a random sample of n subjects, explain why it is about 95% likely that the sample proportion has an error of ne more than 1/n as an estimate of the population proportion (Hint: To show this "In Rule." find two standard enors when 7 = .5. and explain how this compares to two standard erions
55 *An article reports that (4.0.5.6) is a 95% confidence interval for the mean length of stay, in days. of patients in hospital for a particular operation. The article reports the sample size of 50, but not the sample mean or standard deviatiou Find them
56. To encourage subjects to make responses on sensitive questions, the method of random- ized response 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,
57. "To construct a large-sample confidence interval for a proportions, it is not necessary to substitute for the unknown value of x in the formula for the standard error of . A less approximate method for constructing a 95% confidence interval finds the endpoints by determining the 7 values that
58. * Derive the formula for determining how large a sample size is needed to estimate a pop- ulation mean to within B units with probability .95.
1 For a large-sample test of H = 0 against H, 0. the test statistic equals 1 04.a) Find the P-value, and interpretb) Suppose -2.50 rather than 1.04 Find the P-value. Does this provide stronger. or weaker, evidence against the null hypothesis? Explain.
2. Complete the previous exercise for the alternatives (i) H.. >0. (ii) H
3. The P-value for a large-sample test about a mean is P = .10.a) If the alternative hypothesis is H. p. report the value of the test statisticb) Does this P-value provide stronger. or weaker. evidence against the null hypothesis than P .010? Explain.
4 Repeat (a) of the previous exercise for (i) H. > Ho, (u) HH < Ho-
5. Find and interpret the P-value for testing Hu=100 against Ho 100, if a sample of 400 observations has s = 40 anda) Y = 103,b) Y=97.
6. Refer to the previous exercise. Repeat parta) ifa) n = 1600 instead of 400,b) s = 20 instead of 40. Comment on the effects of n and s on the results of a significance test.
7. 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 the 996 responses in the 1991 General Social Survey. With scoles (2.1,-1,-2) for the four
8. The mean score for all high school seniors taking a college entrance exam equals 500. A study is conducted to see whether a different mean applies to those students born m a foreign country. For a random sample of 100 of such students. the mean and standard deviation on this exam equal 508 and
9. A social psychologist plans to conduct an experiment with a random sample of 49 children from some school district Before conducting the experiment, the psychologist checks how this sample compares to national norms on several variables that could affect the re- sults of the experiment. The IQ
10. Refer to Example 6.2. Suppose we instead use the scores (-3, -2. -1, 0, 1, 2, 3), sub- tracting 4 froin each of the original scores. We then test Ho p=0. Explain the effect of the change in scores ona) the sample incan and standard deviation.b) the test statistic,c) the P-value and
11. The inean age at first marnage for inarried men in a New England community was 280 in 1790. For a random sample of 40 married men in that community in 1990, the sample mcan and standard deviation of age at first marriage were 26.0 and s = 9.0.a) State the hypotheses, and find the test statistic
12 A joint USA Today/CNN/Gallup Poll in July 1995 indicated that of 832 white adults. 53% thought affirmative action has heen good for the country and 37% thought it had not been good: the remaining 10% were undecided. Let denote the population proportion of white adults who believe that
13. An experiment consists of giving everyone in some group a lottery ticket, and then later asking if they would be willing to exchange their ticket for another one, plus a small inonc- tary incentive. Let denote the population proportion who would agree to the exchange When this experiment was
14. Let x denote the proportion of Floridians who think that government environmental reg- ulations are too strict Test Ho: r = .5 against H: 5 using data from a telephone poll of 834 people conducted in June 1995 by the Institute for Public Opinion Research at Florida International University, in
15. The 1994 General Social Survey asked. "Do you think it should be possible for a pregnant woman to obtain a legal abortion if the family has a very low income and cannot afford any more children"" Let z denote the population proportion who would answer yes to this question. An anti-abortion
16. A mayoral election in Madison, Wisconsin, has two candidates. Exactly half the residents of the city currently prefer each candidate.a) Find the probability that, for a random sample of 400 residents. at least 230 indicate a preference for one of the candidates or the other? This is also the
17. 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 be- gan 6% of her sentences with the word whereas. To test the histonan's hypothesis, it is decided to count the number of sentences in the disputed document that begin with the word whereas Out of the 300 sentences in the document, none begin with that word Let denote the
18. Refer to the ordinal scale in Table 6.2. Treat this as a qualitative variable, as follows. Ig- nore the subjects responding in the moderate category. Of the remainder not responding in that category, let denote the population proportion who make one of the three lib- cial responses and let 1 -
19. The owner of a department store in Rochester, New York, initiates a week-long newspaper advertising campaign to increase awareness of the store. Before investing any additional money in advertising, the owner takes a phone survey to check whether potential cus- tomers are more inclined or less
20 A multiple-choice test question has four possible responses. The question is designed to be very difficult, with none of the four responses being obviously wrong, yet with only one correct answer It first occurs on an exain taken by 400 students. The designers test whether more people answer the
21. Two researchers conduct separate studies to test Ho=500 against Hop 500. The first researcher gets 519.5, with a standard error of 10.0. The second researcher gets =519.7. with a standard error of 10.0.a) Using a .05, test in each case whether the result is "statistically significant "b)
22. A study considers whether the mean score on a college entrance exam for students in 1996 is any different from the mean score of 500 for students who took the same cxam in 1966. Let / represent the mean score for all students who took the exam in 1996. Test 11 = 500 against H. 500, if for a
23 Report the 1-score that multiplics by the standard error to form aa) 95% confidence interval with 5 observationsb) 95% confidence interval with 15 observationsc) 95% confidence interval with 25 obscrvations.d) 95% confidence interval with df = 25.e) 99% confidence interval with df = 25.
24. Ar test for a mean uses a sample of 15 observations Find the a P-value of P=05 when the alternative hypothesis isa) H.c) Ha: ft < 0? test statistic value that has > 0,b) H: 0,
25. A school board commissions a study of the absentee rate of students at a particular high school in Vancouver For a random sample of 20 student records from the previous year, the mumber of days absent in the previous year is recorded, yielding = 40 and s = 4.a) Stating the necessary
26. Refer to Example 6.11. For the 15 feinale infants, Table 6.9 shows an SPSS printout for the data on the change in heart rate.a) Explain how to interpret all the results in this table.b) Explain how to test whether the true mean is 0. Report the P-value for (i) a one-sided alternative of a
27. Explain how the sample size affects the width of a small-sample confidence interval through its effect on thea) standard error,b) r-score that multiplies the standard error.
28. According to a union agreement, the mean income for all senior-level assembly-line work- cis in a large company equals $500 per week. A representative of a women's group de- cides to analyze whether the mean income for female employees matches this nonn For a random sample of nine female
29. Refer to the previous problema) For which a-levels in Table B (the distribution) can you reject H?b) For which confidence coefficients would the confidence interval contain 500c) Use (a) and (b) to illustrate the correspondence between results of tesis and results of confidence intervals
30 By law. an industrial plant can discharge no more than 500 gallons of waste water per hour, on the average, into a neighbonng lake. Based on other infractions they have noticed, an environmental action group believes this limit is being exceeded. Monitoring the plant is expensive. and only a
31 A study was conducted of the effects of a special class designed to improve children's verbal skills. Each child took a verbal skills test twice. both before and after a three-week period in the class. Let Y be the second exam score minus the first exam score. Ilence, if (the population mean for
32 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 .53 If a jury of size 12 is selected at random from the list of available jurors.a) Find the probability that no women are selected.b) Find the
33. Refer to the previous problem. Test the hypothesis that the selections are random, if no woman is selected out of a sample of size 12. Report the P-value. and interpret.
34 At current rates, the proportion of deaths of Amencan females that are due to suicide is 01 (Statistical Abstract of the United States, 1995). For a random sample of ten deaths. hind the probability thata) None of them is due to suicide.b) At most oue is due to suicide.
35. Refer to the previous problem The proportion of deaths due to suicide for Amencan males is 02.a) For 1000 American male deaths, find the mean and the standard deviation of the prob- ability distribution of the number that were due to suicide.b) Would it be surprising if uone were due to
36. For each free throw, a basketball player has probability .80 of making the shot and 20 of missing it.a) Find the probability that he makes ten free throws in a rowb) He is fouled and takes two free throws. Find the probabilities in the distribution of the number of free throws that he makesc)
37 A football team has probability 50 of winning any particular game.a) Find the probability that the team wins all or none of its six conference games. What assumptions does this calculation require?b) If the team wins all six games, can you be quite confident that its actual probability of
38 A person claiming to possess extrasensory perception (ESP) says she can guess more of- ten than not the outcome of a flip of a balanced coin in another rooin. not visible to hera) Introduce appropnate notation, and state hypotheses for testing her claimb) Of ten coin flips. she guesses the
39 A weather forecaster states "The chance of rain is 50% on Saturday and 50% again on Sunday So. there's a 100% chance of rain sometime over the weekend." If whether it Lains on Saturday is independent of whether it rains on Sunday. find the actual probability of rain at least once during the
40. * Refer to Table 6.2. Of those subjects responding in one of the two extreme categories. letx denote the population proportion who make the extremely liberal response, with 1-being the population proportion making the extremely conservative response. Us- ing small-sample methods, show that the
41. In a given year: the probability that an Aincrican female adult dies in a motor vehicle accident equals .0001 (Statistical Abstract of the United States, 1995).a) In a city having I mullion American female adults, find the mean and standard deviation of the number of deaths from motor vehicle
42 Refer to Problem 6.16. Find the P-value for testing Ho: x = 5 against He. x + 5 when five people are randomly selected and all of them prefer a particular candidate Interprot For these hypotheses. show that a sample of at least six people is
needed to obtain P-value below .05.
43. A fraternal organization admits 80% of all applicants who satisfy certain requirements. Of four members of a minority group who recently applied for admission, all met the re- quirements but none was accepted.a) Find the probability that none would be accepted if the same admissions standards
44. A decision is planned in a test of Ho. =0 against H: >0, using = .05. If = 5, the probability of a Type II error equals .17.a) Explain the meaning of this last sentence.b) If the test were conducted at the a = 01 level, would the probability of a Type II enor be less than. cqual to, or greater
45 Refer to Example 6 11.a) Show that the probability of Type 11 error is less than .001 at = 15.b) Show that the probability of Type I error equals .49 at = 5c) Find the probability of Type II error al u = 8.d) Ifo = .01 instead of 05. venify that the probability of Type II error at = 10 cquals
46. Refer to the hypotheses in Example 6.11. Suppose a sample of size 25 were taken (in- stead of 36) and suppose the estimated standard error of Y is o = 3.6. When a = .05. compute the probabilities of a Type II error for the altcmative u values of (a) 5. (b) 10, (c) 15. Compare these to the
47 Let denote the proportion of schizophrenics who respond positively to a particular treat- ment A test is conducted of 15 against H: >.5. based on a sample of size 25. using a = .05.a) Find the region of sample proportion values for which He is injected. for a large-sample z lestb) Suppose that,
48. Refer to the WWW data set (Problem 1.7).a) Test whether the mean political ideology differs from 4.0 Report the P-value, and interpretb) Test whether the proportion lan onng legalized abortion equals. or differs, from .50. Report the P-value, and interpret
49. Refer to the data file you created in Problein 1.7. For variables chosen by your instructor. conduct inferential statistical analyses Prepare a report. summarizing and interpreting your findings In this report, also use graphical and numerical methods presented earlier in this text to describe
50. In response to the question "Do you think it should or should not be the government's responsibility to provide a job for everyone who wants one." the categories "Definitely should be," "Probably should be." "Probably should not be." "Definitely should not be had response counts 242, 330, 333,
51. An article in a sociology journal that deals with changes in religious beliefs over time states. "For these subjects. the difference in their responses on the scale of religiosity be- tween age 16 and the current survey was significant (P < .05)."a) Explain what it means for the result to be
52. A newly created random number generator is supposed to gencrate a sequence of digits such that each digit is equally likely to be any of 0, 1, 2,... 9. The first 20 numbers gen- erated are 7,7.3, 0.5.6, 3, 2, 6. 1, 0. 9. 9. 4, 0, 8. 5, 0.6.2 As a check of whether the process works correctly.
53. In making a decision in a significance test, a researches worries about the possibility of re- jecting the null hypothesis when it is actually true Explain how to control the probability of this type of error.
54 Consider the analogy between making a decision about a hypothesis in a significance test and making a decision about the innocence or guilt of a defendant in a criminal trial. Iden- tify "Ho true." "Ho false" with "Defendant innocent. "Defendant guilty." and "Reject Ho." "Do not reject Ho" with
55. Medical tests for diagnosing conditions such as cancer or HIV+ are fallible. just like de- cisions in significance tests Identify (Ho true. Ho false) with (Disease absent. Disease present). and (Reject Ho. Do not reject Ho) with (Diagnostic test is positive, Diagnostic test is negative), where
56. An article in a political science journal states that "no significant difference was found between men and wonen in their voting rates (P = .63). Can we conclude that the pop- ulation voting rates are identical for men and women? Explain.
57. A research study conducts 60 significance tests. Of these. 3 are significant at the 05 level. The authors write a report stressing only the three cases in which they found "significant" results, not mentioning the other 57 tests they conducted that were "not significant." Ex- plain what is
58. Some medical journals have a policy of publishing results only of research that is statis- tically significant at some levela) Explain the dangers of this hy describing what could happen if researchers at 20 differ- ent institutions independently and without communication ainong each other,
59 A 95% confidence interval lor is (96. 110) Which of the following statements about significance tests for the saine dana is tare) correct?a) In testing Ho 100 against H:b) In testing H100 against H,c) In testing Ho: = against H: the confidence interval. 100, P > .05. 100, P < .05. 1. P>05 if is
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