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Statistical Methods For The Social Sciences 3rd Edition Alan Agresti, Barbara Finlay - Solutions

24. For college freshmen in 1994. the odds ratio between gender (male, female) and opinion about whether homosexual relations should be legally prohibited (yes, no) equaled 2.6 (Higher Education Research Institute. University of California at Los Angeles)a) Explain what is wrong with the
25. Consider the 3 x 3 table having entries, by row, of (4.2.1/2, 2, 2/1, 2, 4).a) Is the chi-squared test of independence appropriate for these data? Explain.b) Using software, conduct a valid test of independence. Interpret.
26 Refer to the previous exercise. Suppose the variables are ordinal. with levels (high. medium, low) for each classification.a) Compute the numbers of concordant and discordant pairsb) Compute gamma, and interpret.c) Show how to express gamma as a difference between two proportions.d) Using
27. Table 8.32 refers to a study that assessed factors associated with women's attitudes toward mammography. The columns refer to their response to the question, "How likely is it that a mammogram could find a new case of breast cancer?" Use gamma to summarize the strength of association. Interpret
28. A study on educational aspirations of high school students (S. Crysdale, International Journal of Comparative Sociology, Vol. 16. 1975, pp. 19-36) measured aspirations us- ing the scale (some high school. high school graduate. some college. college graduate). For students whose family income
29 Table 8.34 comes from a study of economic conditions and political attitudes of Cuban workers toward the Cuban revolution. A compute: analysis reports: Statistic Value Gamma -0.481 Kendall's Tau-b -0.238a) Interpret one of these sample statistics. ASE 0.117 0.059b) Obtain a 95% confidence
30. Refer to Problem 8.10. The analysis there does not take into account the ordinality of the vanables. Using software, summarize the strength of association by finding and interpret- ing gamma. Construct and interpret a 95% confidence interval for the population value of gamma.
31. Suppose .30 for the relationship between two cadinal variables.a) Of the pairs that are concordant or discordant, what proportion are concordant? Dis- cordant'b) Is this a suonger or a weaker association than one having =-70? Explain
32. Conduct the test of independence for Table 8.17 using Kendall's tau-b rather than gamma. Compare results to those obtained using gamma
33. Refer to the WWW data set (Problem 1.7). Using computer software. create and analyze descriptively and inferentially the contingency table relatinga) Political affiliation and opinion about abortion.b) Religiosity and opinion about abortion.c) Gender and belief in life after death
34. Refer to the data bile you created in Problem 1.7 For variables chosen by your instruc- to1. conduct descriptive and inferential statistical analyses. Interpret and summarize your findings.
35. Several sociologists have reported that racial prejudice varies according to religious group Examine this using Table 8.35, for white respondents to the 1980 General Social Sur- vey. Religious preference has four categories: (i) Liberal Protestant, encompassing the theologically more liberal
36 Analyze the data in Table 8.36, relating political ideology to political party Prepare a report of your analyses, emphasizing interpretations of the results.
37. Table 8.37 classifies a sample of psychiatric patients by their diagnosis and by whether their treatment prescribed drugs. Analyze these data. providing interpretations of all your analyses.TABLE 8.37 Drugs TABLE 8.38 Belief In Afterlife Diagnosis Yes No Gender Yes No. Undecided Schizophrenia
38. Table 8.38, from the 1991 General Social Survey, refers to the association between belief in the afterlife and gender. Analyze these data.
39 The printour in Table 8.39 refer to a sample of voters from a recent presidential primary in Wisconsin. Democrats and Republicans were classified on political ideology. Explain how to interpret these analyses TABLE 8.39 PARTY IDEOLOGY Frequency Expected Conserv |Moderate Liberal Total --+----
40 The sample in Table 8.17 consists of 104 black Americans. A similar table relating in- come and job satisfaction for white subjects in the 1991 General Social Survey has counts (18. 39, 36) in row 1, (43. 163, 158) in row 2, and (27, 97, 156) in row 3. Analyze these data.
41. Refer to Problem 8.8. A similar table relating alcohol use as rows to manjuana use as columns has cell counts (322, 5) in row 1 and (994, 955) in row 2. Analyze these data.
42. In a survey conducted in 1994 by Market Segment Research and Consulting, Inc., as part of the 1994 Ethnic Market Report, ahour 5000 respondents were asked "Which of the fol- lowing issues facing your community today do you feel is the most important?" Table 8.40 shows the data. Describe the
43. Shortly before a gubernatorial election, a random sample of 50 potential voters are asked the following questions Do you consider yourself to be a Democrat (D), a Republican (R), or Independent (I)? If you were to vote today, would you vote for the Democratic candidate (D), the Republican (R).
44 Give an example of a contingency table for which the chi-squared test of independence should not be used. because ofa) Sample sizeb) Measurement scalec) The rows of the table are dependent samples
45. True or false? Interchanging two rows in a contingency table has no effect on the chi- squared statistic.
46 True or false? The null hypothesis for the test of independence between two categorical variables is Hex2=0
47.a) When the sample size is very large, we have not necessarily established an important result when we show a statistically significant association. Explainb) The remark in Section 84 about small P-values not necessarily referring to an impor- tant effect applies for any significance test.
48. True or false? If y = 0 for two variables, then the variables are statistically independent Explain.
49. True or false? Interchanging two rows in a conungency table has no effect on gamma.
50. The correct answer in Problem 8.45 implies that if the chi-squared statistic is used for a contingency table having ordered categones in both duections. then (select the correct response(s)):a) The statistic actually treats the variables as nominal.b) Information about the ordering is
51 Show how to get the chi-squared values for df = 1 in Table C from z-scores in the stand- ard normal table (Table A). Illustrate for the chi-squared value that has P-value 01.
52. Each subject in a sample of 100 men and 100 women is asked to indicate which of the fol- lowing factors (one or more) are responsible for increases in crime committed by teenagers A - the increasing gap in income between the nch and poor, B - the increase in the per- centage of single-parent
53. * Tahle 8.43 exhibits the maximum possible association between two binary variables for a sample of size n. Show that x= for this table and, hence, that the maximum value of x for 2 x 2 tables is n.
54. Refer to the previous exercise. The phi-squared measure of association for 2 2 contin- gency tables has sample value Explain why this measure falls between 0 and 1, with a population value of 0 correspond- ing to independence. (It is a special case, for 2x2 tables, of the Goodman and Kruskal
55 For 2 x 2 tables. gamma simplifies to a measure first proposed about 1900 by the statis- tician G. Udny Yule, who also introduced the odds ratio. In that special case, gamma is called Yule's Q.a) Show that for a genenc table with counts (a.b) in row 1 and (c.d) in row 2, the number of concordant
56 You have data on two ordinal variables, degree of political conservatism (low, niedium, high) and willingness to raise taxes for welfare programs (low, medium. high). If you compute gamma for these data, would you expect its value to be positive, or would yon expect it to be negative? Explain.
57. * Construct a 3 x 3 table for cach of the following conditions:a) Gamma equals 1. (Hint: There should be no discordant pairs.)b) Gamma equals -1.c) Gamma equals 0.
58. * Goodman and Kruskal's tan, also called the concentration measure, is a PRE measure of association for nominal vanables. We illustrate it for Table 8.35. Rule 1 for predicting attitude without using religious preference is based on the marginal distribution for that response variable. To
59 Refer to the previous problem. A contingency table relating race (the rows) to opinion (the columns) has counts (490, 410) in row 1 and (10.90) in row 2. A contingency table relating gender (the rows) to opinion has counts (350, 150) in row I and (150.350) in row 2.a) Compute the difference of
60. * Another PRE measure of association for nominal variables is called lambda. For this measure, rule 1 predicts for each observation the modal category for the marginal distri- bution of the response variable. For Table 8 35, for example, rule 1 predicts Oppose al- ways. Rule 2 predicts the
1. For the following variables in a regression analysis, which variable more naturally plays the role of X (explanatory variable) and which plays the role of Y (response variable)"a) College grade point average (GPA) and high school GPAb) Number of children and mother's education level.c) Annual
2 Sketch plots of the following lincs. for values of X between 0 and 10:a) y = 7+.5Xb) y=7+Xc) Y=7-Xd) Y = 7-.5Xe) ' = 7f) y = X
3 For the data on the 50 states in Table 91 on Y = violent crime rate and X = poverty rate. the prediction equation is =209.9+ 25.5x.a) Sketch a plot of the prediction equation for X between 0 and 100.b) Interpret the Y-intercept and the slope.c) Find the predicted violent crime rate for
4. A college admissions officer claims that the prediction equation = .5+7.0X approxi- mates the relationship between Y = college GPA and X = high school GPA (both mcas- ured on a four-point scale) for students at that college.a) Is this equation realistic? Why or why notb) Suppose that the
5. A recent study of mail survey response rate patterns of the elderly found a prediction equa- tion relating X = age and Y = percentage of subjects responding of = 90.2-.6X, for ages between about 60 and 90 (D. Kaldenberg et al. Public Opinion Quarterly, Vol 58, 1994, p. 68).a) Interpict the
6. Refer to Problem 9.24. For those counties, Table 9.9 shows part of the printout for the regression analysis relating Y = median income (thousand of dollars) to X = percent of residents with at least a high school education.a) Report the prediction equation, and interpret the slopeb) Is the
7. A high school student analyzes whether a relationship exists between X = number of books read for pleasure in the previous year and Y = daily average number of hours spent watching television. For her three best friends. the observations are as shown in Table 9.10.a) Construct a scatter diagram.
8 For the WWW data set described in Problem 1.7, the sample correlation between Y = political ideology (scored 1 to 7) and X = mumber of times a week reading a newspaper is r = -.066a) Interpret the sign of the correlationb) Interpret the square of the correlation. Would you conclude that the
9. The prediction equation for a sample of 100 people relating X = years of education and Y = annual income (in dollars; is = -6000 + 3000X, and the Pearson correlation equals .50.a) Suppose instead that Y refers to annual income, in thousands of dollars. State the pre- diction equation and the
10. For the house sales data in Table 9.4. Table 9.11 shows a computer printout for the regres- sion analysis relating selling price (thousands of dollars) to number of bedrooms TABLE 9.11 Variable PRICE BEDROOMS N Mean Std Dev 93 99.5333 44.1841 93 3.1828 0.6070 Sum of Mean Source DE Squares Model
11. Refer to Table 9.1. Table 9.12 shows an SPSS printout for the relationship for all 51 ob- servations between Y = murder rate and X-percentage whitea) Report the prediction equation. Interpret the Y-intercept and slopeb) Report the coefficient of determination, and interpret.TABLE 9.12 R Square
12 Refer to Table 9.1 For all 51 observations, use software to analyze the relationship be- tween murder rate and violent crime rate, treating murder rate as the response variable.a) Construct a scatter diagram. Does there seem to be a positive, on a negative. relation- ship?b) Find the prediction
13 Refer to Problem 9 24. For those data, use software to analyze Y = cnme rate and X = percentage living in an urban environment.a) Construct a stem and leaf plot and a box plot for Y Interpret.b) Show that 24 5+.56X. Interpret the Y-intercept and slope.c) Find the predicted crime rate and the
14. Using software, plot the relationship between percentage single-parent families and per- centage white, for the data in Table 9.1.a) Based on your plot, identify the two observations that seem quite different from the othersb) Find the prediction equation and the correlation (i) for the entire
15 Refer to the housing data in Table 94. Use software to fit the model with number of bath- roorns as the predictor for price.a) Construct a scatter diagram. Note the effect of using such a highly discicte predictor. having only three values.b) Find the prediction equation. Interpret the slope.c)
16 A study was conducted using 49 Catholic female undergraduates at Texas A & M Univci- sity. The variables measured refer to the parents of these students. The response variable is the number of children that the parents have One of the explanatory variables is the mother's educational level.
17 Table 9.13 lists recent values for several nations on the crude birth rate (number of births per 1000 population size) women's economic activity (female labor force as percentage of male). percentage women using contraception, female life expectancy, female adult literacy rate, a human
18. Refer to the previous exercise. Now use GNP as the explanatory variable for predicting birth rate.a) Construct a stem and leaf plot or box plot for birth rate, and describe its distribution.b) Construct a scatter diagram, and indicate whether a linear model seems appropriate.c) Fit the model,
19. Refer to the previous two exercises. Using software, obtain the correlation matrix for these data. Which pairs of variables are highly correlated? Describe the nature of those correlations, and explain how your software handled the missing values. (For a partic- ular analysis. most software
20. For a random sample of U S. counties, data are obtained on X = percentage of the popo- lation aged over 50 and Y = per student expenditure on education. Table 9.15 is part of the computer printout for the analysis.a) What was the sample size for this study?b) Fill in the blanks in Table 9.15.
21 For Table 9.1, use software to analyze the data on violent crime rate and percent single parent families.a) Construct a scatter diagram. What does it show"b) One point is quite far removed from the others. having a much higher value on both variables than the rest of the sample, but it fits in
22. Refer to the WWW data set (Problen 1.7) Using softu are, conduct regression analyses relating (i) political ideology and X = religiosity, (ii) Y = high school GPA and Xhours of TV watching. Prepare a report.a) Using graphical ways of portraying the individual variables and their relationshipb)
23. Refer to the data file you created in Problem 17. For variables chosen by your instruc- tor, conduct a regression and correlation analysis. Report both descriptive and inferential statistical analyses, interpreting and summarizing your findings.
24. Table 9.16 shows recent data from all 67 Florida counties on crime rate (number of crimes per 1000 residents). median income (in thousands of dollars), percentage of residents with at least a high school education (of those aged at least 25). and the percentage of the county's residents living
25 Refer to the previous exercise. Using income as the response vanable and percentage of high school graduates as the explanatory variable, analyze these data Prepare a report. and explain carefully the interpretations of all your analyses
26. Refer to Table 9.1. Analyze the relationship between violent crime rate and percentage having at least a high school education Write a report showing your analyses, providing interpretations, and summarizing your findings.
27 Refer to Table 9.1. Analyze the relationship between violent crime rate and percentage of single-parent families. Write a report showing your analyses, providing interpretations. and summarizing your findings.
28 Repeat the previous exercise, using murder rate as the response variable.
29. Refer to Table 9.13. Analyze the relationship between new spaper circulation and gross national product Tell why you conducted each part of the analysis and explain how 10 interpret the results
30 Recently the General Social Survey has asked subjects to rate various groups using the "feeling thermometer." Ratings between 50 and 100 mean you feel favorable and warm Toward the group, whereas ratings between 0 and 50 mean that you don't feel favorable. It also asks subjects to rate
31. In an article in USA Today (December 28, 1984), sociologists N. Glenn and B. A Shel- ton are quoted as showing a strong link between residential mobility and divorce rates In Table 9.18, divorce rate is the annual number of divorces and annulments per 1000 pop- ulation, and mobility rate is the
32. Describe a situation in which it is inappropriate to use the Pearson correlation to measure the association between two quantirative variables.TABLE 9.18 Region Mobility Divorce Rate Rate New England 41 4.0 Middle Atlantic 37 3.4 East North Central 44 51 West North Central 46 4.6 South Atlantic
33. Annual income, in dollars. is the response variable in a regression analysis. For a British version of the report on the analysis, all responses are converted to British pounds sterling (1 pound equals about 1.5 dollars. as of 1997).a) Ilow, if at all. does the slope of the prediction equation
34. The variables annual income (thousands of dollars). X = number of years of edu- cation, and X2 number of years experience in job are measured for all the employees having city-funded jobs, in Knoxville, Tennessee. The following prediction equations and correlations apply i. = 10+ 1.0X;. ii. =
35. Que can interpret r = .3 as follows.a) A 30% reduction in error occurs in using X to predict Y.b) A 9% reduction in error occurs in using X to predict Y compared to using to predict Y.c) 9% of the time Yd) Y changes .3 unit for every one-unit increase in X.e) When X predicts y, the average
36. The conclation is inappropriate as a measure of association between two quantitative vari- ables.a) When different people measure the variables using different units.b) When the relationship is highly nonlinear.c) When the data points fall exactly on a straight lined) When the slope of the
37 The slope of the least squares prediction equation and the Pearson correlation coefficient are similar in the sense thata) They do not depend on the units of measurementb) They both must fall between-1 and +1.c) They both have the saine sign.d) They both equal when there is the suongest
38. Describe the assumptions (a) in using the regression equation E(Y)=-8X to represent the relationship between two vanables and (b) in inaking inferences about that equation using the least squares prediction equation. Which assumptions are most critical?
39. Refer to the previous exercise In view of these assumptions. indicate winy such a model would or would not be good in the following situationsa) X = time, percentage unemployed workers in the United States. (Hiru Docs this continually tend to increase or decrease?}b) X = income, y = charitable
40 For a class of 100 students, the teacher takes the 10 students who perform poorest on the midterm exam and enrolls theun in a special tutoring program. The overall class mean is 70 both on the midterm and final, but the mean for the specially tutored students increases from 50 to 60. Can we
41 Refer to Problein 9 24 For these counties, the correlation between high school educa- tion rate and income equals 79. Suppose we also have data at the individual level as well as aggregated for a county. Sketch a scatter diagram to show that at the individual level. the correlation could be
42. Explain why the correlation between X = number of years of education and y = annual income is likely to be smaller if we use a random sample of adults who have a college degree than if we use a random sample of all adults.
43 Explain carefully the interpretations of the standard deviations (a! Sr. (b) sx. (c) 8.
44 *A report summarizing the results of a study on the relationship between scores for stu- dents on a verbal aptitude test X and a inathematics aptitude test I' states that X = 480. =500, sx=80, sy = 120, and r = .60.a) Using the formulas for the correlation and for the least squares estimates,
45 *Observations on both X and Y are standa dized, having estimated means of 0 and stand- and deviations of 1 (see Section 42) Show that the prediction equation has the form =rX. where r is the sample correlation between X and Y, that is, for the standardized variables, the Y-intercept equals 0 and
46 *A confidence interval for a population correlation p requires a inathenatical transfor- ination of r for which the sampling distribution is approximately normal This transfor- mation is T(r) = (1/2) log[(1+r)/(1r)], where log denotes the natural (base-e) loganthm. The transformation of the
47. *Refer to the previous exercise and to Problem 9 16 Find and interpret 95% confidence intervals for ilie population Pearson correlation and the population coefficient of detenni- nation
48. *Refer to Problein 9.46. Let p and denote the population correlation values between two vanables for two separate populations. Let 71 and r denote sample values for inde-pendent random samples from the populations To test the null hypothesis Ho p = p. the test statistic is 72-7 OT:-T where Ti
49. Show that substituting X = X into the prediction equation = a + bx yicids the predicted Y-value of = Y. (Hint: The least squares formula for the Y-intercept is a bx) Show that this means that the least squares prediction equation passes through the point with coordinates (X.). the center of
50. Alternative formulas for defining the Pearson correlation use the data in formulas similai to the one for b: - 11 Roughly, the correlation is the average cross-product of the -score for X times the 2- score for Using this formula, explain why (a) the correlation has the same value when X
51 The values of Y are multiplied by a constantc. From their formulas, show that the stand- ard deviation sy and the least squares slope b are also then inultiplied byc. Thus, show that bsx, sy remains the same. so that r does not depend on the units of measurement. ===
52 'Suppose that the linear regression equation E(Y)=a+BX with normality and constant standard deviation a is truly appropriate for the relationship between Y and X Then, the interval of numbers 1 + "1 (X-X) (X-X) predicts where a new observation on Y will fall at that value of X. This interval is
53. *Refer to Problem 9.16 and the previous exercisea) Construct a 95% confidence inter al for the mean number of children for mothers hav- ing X = 16.b) Explain why the prediction interval is probably inappropriate. (Hint. Is Y approxi- inately normal?)
54 *Refer to Example 9.8 on regression toward the mean. Does this imply that, over time. variation in height is decreasing until, eventually, everyone has the same height? Explain.
55. *To implement least squares, one can find the formulas for the a and b estimates that min- imize SSE (Y- = [[(a + bx)] using calculus. by taking the denvative of this function with respect toa, taking the derivative with respect tob. setting the two derivatives equal to 0, and solving the two
1 For each of the three criteria for a causal relationship. describe a relationship between two variables that is not causal because that criterion would be violated
2. An association exists between college GPA and whether one has ever used marijuanaa) Explain how the direction of a causal arrow might go in either direction.b) Explain how a third vanable might be responsible for the association.
3. Give an example of two variables for which the time order for a causal connection is am- biguous, and explain why.
4. Explain clearly what it means to control for a vanable. Describe a situation in which an association would change dramatically after controlling a certain variable
5 Explain what is meant by a spurious association. = ==a) Illustrate using the variables X = shoe size. X2 = age, and number of books one has ever read. for a random sample of children froin schools in Winnipeg, Canada.b) Illustrate using the vanables X, shoe size. X2 gender, and Y = annual income,
6. Refer to Problem 9.17 (Table 9 13). Those data exhibit a strong negative correlation be- tween birth rate and per capita television ownership. While there could potentially be a causal relationship, there are other plausible explanations for this association. Using another variable from that
7 Table 10.9 iclates occupational level (white collar, blue collar) and political party choice. controlling for income.a) Construct the bivariate table between occupational level and political party, ignoring income is there an association? If so, describe it.b) Do the partial tables display an
8. In murder inals in 20 Florida counties during 1976 and 1977, the death penalty was given in 19 out of 151 cases in which a white killed a white, in 0 out of 9 cases in which a white killed a black. in 11 out of 63 cases in which a black killed a white, and in 6 out of 103 cases in which a black

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