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

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(Slutsky’s lemma) Let {Xn, n = 0, 1, 2 . . . } and {Yn, n = 0, 1, 2 . . . } be sequences of random variables. If Xn d−→ X0 and Yn p−→ c for a constant c as n → ∞, then show that(1) Xn + Yn d−→ X0 + c.(2) XnYn d−→ cX0.(3) Xn/Yn d−→ X0/c, (c ̸= 0).
Suppose that X0,X1, . . . is a sequence of random variables defined on the probability space (Ω,A, P). If g : (R, B1) → (R, B1) is a continuous function, then show that(1) If Xn−a−.s→. X0 then g(Xn) −a−.s→. g(X0).(2) If Xn p−→ X0 then g(Xn) p−→ g(X0).(3) If Xn d−→ X0 then
Prove the statements of (i)- (v) of Theorem 2.6 for the continuous case.
Prove the statements of (i)- (iii) of Theorem 2.6 for the discrete case.
Let E|X|n < ∞ for some positive integer n. Show that the nth derivative of ϕX(t) exists and is continuous on R, andϕ(n)X (t) ≡ dn dtn ϕX(t) = E©(iX)neitXª, (2.75)and in particular, E(Xn) =1 in ϕ(n)X (0). (2.76)
Let X1, . . . ,Xn be independent random variables, and let Sn = X1 +· · ·+Xn. Show that the characteristic function of Sn is the product of the characteristic functions of the Xi.
Compute the expectation, variance and characteristic function of each distribution in Table 2.1.
Prove the statements (i)-(v) of Theorem 2.4.
Let X and Y be independent random variables with E|X| < ∞ and E|Y |
Show that the density (2.25) is symmetric about zero and unimodal.
(i) Let X1, X2 be independent, each distributed according to the Cauchy distribution (2.24). Find the distribution of a1X1 + a2X2. (ii) Use (i) to prove (by induction) that if X1, . . . ,Xn are independent, each distributed according to (2.24), then Sn is also distributed according to (2.24).
Let the distribution of X be N(μ, σ2) and Y = aX +b. Show that the distribution of Y is N(aμ +b, a2σ2).
Show that we can replace intervals of the form (a, b] in the definition of the family of Borel sets by other classes of intervals, for instance, all closed intervals, all intervals [a, b),a, b ∈ R, all intervals (a,∞), a ∈ R, all intervals [a,∞), a ∈ R, all intervals (−∞, b), b ∈ R,
Suppose that B is a σ-field of Ω and A is a nonempty subset of Ω. Show that the collection of events B ∩ A ≡ {B ∩ A : B ∈ B} (2.73)is a σ-field of A.
Let A ⊂ Ω. Show that σ[A] ≡ {∅, A,Ac,Ω} is a σ-field of Ω.
We denote the collection of all subsets of Ω by S(Ω). Show that S(Ω) is aσ-field of Ω.
Let A be a σ-field, and An ∈ A for all n ∈ N. Show that the following statements hold:(B4) ∅ ∈ A.(B5)Sn i=1 Ai ∈ A.(B6)Tn i=1 Ai ∈ A.(B7)T∞i=1 Ai ∈ A.(B8) lim supk→∞ Ak ∈ A.(B9) lim infk→∞ Ak ∈ A.Here, we define lim supk→∞ Ak ≡T∞n=1 S∞k=n Ak and lim infk→∞
Show that A and ()-(4)-04 (2.72)
13.32. Suppose we use a centered variable for the covariate and express the interaction model when the categorical predictor has two categories as E{y) = a + piix - ixx) + P2Z + /^(x - ^)Xz.Explain how to interpret P2. and explain how this differs from the interpretation for the model without a
13.30. In the United States, the mean annual income for blacks (p.]) issmallerthan for whites {p.2), the mean number of years of education is smaller for blacks than for whites, and annual income is positively related to number of years of education. Assuming that there is no interaction, the
13.29. In the model E{y) = a + /3ix + /ibz, where z is a dummy variable,(a) The categorical predictor has two categories.(b) One line has slope /3i and the other has slope Pi-(c) (S2 is the difference between the mean ofy for the second and first categories of the categorical variable.(d) @2 is the
13.28. A regression model is fitted to annual income(thousands of dollars), using predictors age and marital status. Table 13.20 shows the sample mean incomes and the adjusted means for the model.How could the adjusted means be so different from the unadjusted means? Draw a sketch to help
13.27. Draw a scatterplot with sets of points representing two groups such that Htf. equal means would be rejected in a one-way ANOVA but not in an analysis of covariance.
13.26. Let y = death rate and x = mean age of residents. measured for each county in Louisiana and in Florida. Sketch a hypothetical scattcrplot, identifying points for each state, when the mean death rate is higher in Florida than in Louisiana when
13.24. You have two groups, and you want to compare their regressions of y on x, in order to test the hypothesis that the true slopes are identical for the two groups. Explain how you can do this using regression modeling.
13.23. For the "2005 statewide crime" data file at the text Web site let z be a dummy variable for whether a state is in the South, with z = 1 for AL, AR, FL, GA, KY, LA, MD. MS, NC, OK, SC, TN, TX, VA, WV.(a) Not including the observation for D.C., analyze the relationship between y = violent
13.22. Analyze the "house selling price 2" data file at the text Web site by modeling selling price in terms of size of house and whether it is new.(a) Fit the model allowing interaction, and test whether the interaction term is needed in the model.(b) Figure 13.14 shows a scatterplot, labeling the
13.20. Table 13.18 is a printout based on GSS data.The response variable is an index of attitudes toward premarital, extramarital, and homosexual sex. Higher scores represent more permissive attitudes. The categorical explanatory variables are race (0 for whites, 1 for blacks), gender (0 for 13.21.
13.19. Table 13.17 shows results of fitting a regression model to data on salaries (in dollars) of about 35,000 college professors. Four predictors are categorical (binary), with dummy variable defined in parentheses. Write a short report, interpreting the effects of the predictors.TABLE 13.17
13.18. An article2 on predicting attitudes toward homosexuality used GSS data to model a response variable with a four-point scale in which homosexual relations were scaled from 1 = always wrong to 4 = never wrong, with xi = education(in years), X2 = age,X3 = political conservative(1 = yes,0 = no),
13.17. Referto the "OECD data" file atthe text Web site, shown in Table 3.11 on page 62. Pose a research question about how GDPand whether a nation isin Europe relates to carbon dioxide emissions. Conduct appropriate analyses to address that question, and prepare a report summarizing your anatyscs
13.13. Refer to the previous exercise. Conduct the analyses for the interaction model and for comparing that model to the no-interaction model, as shown in Sections 13.3 and 13.4, after adding these data.13.14. Exercise 13.1 reported the regression equation relating y = education to race (z = 1 for
13.12. Table 13.1 did not report the observations for ten Asian Americans. Their (x._y) values were Subject 123456789 10 Education 16 14 12 18 13 12 16 16 14 10 Income 70 42 24 56 32 38 58 82 36 20 Conduct the analyses for the no-interaction model shown in Sections 13.2 and 13.4, after adding these
13.11. Refer to the previous exercise. The means of percentage registered for the three categories are x\ - 76.2, *2 - 49.5, J3 = 39.7. The overall mean x = 60.4.(a) Find the adjusted mean on percentage voting for Anglos. Compare it to the unadjusted mean of 52.3, and interpret.(b) Sketch a plot of
13.9. Replicate all the analyses shown in Sections 13.2-13.4 using the "Income, education, and racial-ethnic status" data file at the text Web site.
13.8. Refer to the previous exercise. Table 13.15 shows results of fitting the model allowing interaction, where new^size refers to the cross-product term.(a) Report the lines relating predicted selling price to size for homes that arc (i) new, (ii)not new.(b) Find the predicted selling price for a
13.7. For the "house selling price" data file at the text Web site, Table 13.14 shows results of modeling y = selling price (in dollars) in terms of size of home (in square feet) and whether the home is new (1 = yes; 0 = no).(a) Report and interpret the prediction equation, and form separate
What would you need to do to test the effect of marital status (all categories at once), controlling for the other variables?
What hypothesis does it test?
13.6. For the survey in the previous exercise, the sample size was 1417.(a) Test the null hypothesis that sex has no effect on the response, controlling for the other predictors. Interpret.(b) Construct a 95% confidence interval for the sex effect, controlling for the other predictors.Interpret.(c)
13.5. Based on a national survey, Table 13.13 shows results of a prediction equation reported for y = alcohol consumption, measured as the number of alcoholic drinks the subject drank during the past month (D. Umberson and M. Chen, American Sociological Review, vol. 59,1994, p. 152).(a) Setting up
13.4. For data1 from 27 automotive plants on y =number of assembly defects per 100 cars and x = time to assemble each vehicle (in hours, falling between 12 and 54), y = 61.3 + 0.35x. Adding z = whether facility is Japanese (1 = yes, 0 = no)to the model, y = 105.0 - 0.78.x - 36.2z.(a) Interpret the
13.3. A regression analysis for the 100th Congress predicted the proportion of each representative's votes on abortion issues thattook the "pro-choice"position (R. Tatalovich and D. Schier, American Politics Quarterly, vol. 21, 1993). The prediction equation was y = 0.350 + Omiid + 0.094/- +
13.2. Table 9.13 on page 294 showed data for several nations on various indices. Let y = percentage who use the Internet, x = per capita GDP (in thousands of dollars), and z = whether the nation is a Western nation (1 = yes, 0 = no).(a) The prediction equation for the effect of z is y = 10.5 +
13.1. The regression equation relating y = education(number of years completed) to race (z = 1 for whites, z = 0 for nonwhitcs) in a certain country is E{y) = 11 + 2z. The regression equation relating education to race and to father's education (x) is£(>0 = 3 + 0.8x - 0.6z.(a) Ignoring father's
• ^asg observations having sample mean y, explain why E(« - y)7(s -1)estimates the variance (T2 jn of the sampling distribution of the y,-values.(b) Using (a), explain why 2«(y/ - y)2/(g - 1)estimates a 2. For the unequalsample size case, replacing n by n,- yields the between-groups estimate.
12.57. This exercise motivates the formula for the between-groups variance estimate in one-way ANOVA. Suppose the sample sizes all equal n and the population means all equaljx. The sampling distribution of each y(- then has mean /i, and variance(r2/n. The sample mean of the y,- values is y.(a)
12.55. You know the sample mean, standard deviation. and sample size for each of three groups.Can you conduct an ANOVA F test comparing the population means, or would you need more information?
12.54. Interaction terms are needed in a two-way ANOVA model when(a) Each pair of variables is associated.(b) Both explanatory variables have significant effects in the model without interaction terms.(c) The difference in means between two categories of one explanatory variable varies greatly
• = /Xg is false(a) The smaller the between-groups variation and the larger the within-groups variation?
12.52. One-way ANOVA provides relatively more evidence that Hq: /X| =
12.50. True or false? Suppose that forsubjects aged under 50, there is little difference in mean annual medical expenses forsmokers and nonsmokcrs, but forsubjects aged over 50 there is a large difference. Then there is no interaction between smoking status and age in their effects on annual
12.49. A random sample of 26 female graduate students at the University of Florida were surveyed about their attitudes toward abortion. Each received a score on abortion attitude according to how many from a list of eight possible reasons for abortion she would accept as a legitimate reason for a
12.48. The 25 women faculty in the humanities division of a college have a mean salary of $66,000, whereas the five in the science division have a mean salary of $80,000. On the other hand, the 20 men in the humanities division have a mean salary of $65,000, and the 30 men in the science division
12.46. The null hypothesis of equality of means for a factor is rejected in a two-way ANOVA. Docs this imply that the hypothesis will be rejected in a oneway ANOVA Ftest if the data are collapsed over the levels of the second variable? Explain.
11.63. The numerator R 2 — r-2 VV| of the squared partial correlation rjX2.Xl gives the increase in the proportion of explained variation from adding X2 to the model. This increment, denoted by is called the squared semipartial correlation. One can use squared semipartial correlations to
11.62. Let R 2, r, denote R 2for the multiple regression model with k explanatory variables. Explain why
11.61. The sample value of R 2tends to overestimate the population value, because the sample data fall closer to the sample prediction equation than to the true population regression equation. This bias is greater if n is small or the number of predictors k is large. A better estimate is adjusted R
11.60. Software reports four types of sums of squares in multiple regression models. The Type I (sometimes called sequential) sum ofsquares represents the variability explained by a variable, controlling for variables previously entered into the model.The Type III (sometimes called partial) sum of
11.59. Suppose the correlation between y and X| equals the multiple correlation between y and xi and X2.What docs this imply about the partial correlation fyxyxil Interpret.
11.58. Which of the following sets of correlations would you expect to yield the highest R 2value? Why?(a) r yx\ = 0.4, r yx2 = 0.4, r X IXl = 0.0(b) r yx\ = 0.4, r yx2 = 0.4, r X\X2 = 0.5(e) r yx\ = 0.4, ryx2 = 0.4, r X\X2 = 1.0
11.57. Whenever xi and X2 are uncorrelated, then R 2for the model ^(y) = a + 0ixi + 02X2 satisfies='yxj + r yx2' t^'s casc' draw a figure that portrays the variability in y, the part of that variability explained by each ofxi and X2, and the total variability explained by both of them together.
11.56. For the models E{y) = a + 0x and E{y) =a + 0ixi + 02X2, express null hypotheses in terms of correlations that are equivalentto the following:(a) Hq: 0 = 0(b) Ho: 0i = @2 = 0(c) Ho: 132 = 0
11.55. Give an example of three variables for which you expect 0 # 0 in the model E{y) = a + 0xi but 01 = 0 in the model ^(y) = a + 0ixi + 02-V2.
11.54. Let y = height, xi = length of right leg. X2 =length ofleft leg. Describe what you expect for the relative sizes of rXiX2, R. and ryX2.Xl.
11.53. Explain the difference in the purposes of the correlation. the multiple correlation, and the partial correlation.
11.52. The F test for comparing a complete model to a reduced model(a) Can be used to test the significance of a single regression parameter in a multiple regression model(b) Can be used to test Ho\ fi\ = = fik = 0 in a multiple regression equation(c) Can be used to test Ho' No interaction, in the
11.51. Ify = 2 + 3xi + 5x2 - 8x3,(a) ryXi < 0(b) ryXyXi < 0(c)r yxyX\JC2 ^ ^(d) Insufficient information to answer.(e) Answers (a), (b), and (c) are all correct.
11.49. Ify = 2 + 3x| + 5x2 - 8x3, then controlling for X2 and X3, the predicted mean change in y when xi is increased from 10 to 20 equals(a) 3 (b)30 (c)0.3 (d) Cannot be given—depends on specific values ofX2 and X3.11.50. Ify = 2 + 3xi + 5x2 - 8x3,(a) The strongest correlation is between y and
11.48. In regression analysis, which ofthe following statements must be false? Why?(a) ryXl = 0.01, ryX2 = -0.75,7? = 0.2(b) The value ofthe residual sum ofsquares, SSE, can increase as we add additional variables to the model.(c) For the model E(y) = a + jSixi, y is significantly related to xj at
11.47. Table 11.22 shows results of fitting various regression modelsto data on _v = college GPA,xi = high school GPA, X2 = mathematics entrance exam score, and X3 = verbal entrance exam score. Indicate which of the following statements arc false.Give a reason for your answer.(a) The correlation
11.46. In Exercise 11.1 on y = college GPA, X] = high school GPA, and X2 = college board score, E{y) = 0.20 + 0.50xi + 0.002x2. True or false:Since /3i = 0.50 is larger than fa = 0.002, this implies that xi has the greater partial effect on y.Explain.
11.45. In the study mentioned in the previous exercise, a separate model did not contain interaction terms.The best predictor of attitudes toward homosexuality was educational level, with an estimated standardized regression coefficient of 0.21. The authors also reported, "Controlling for other
11.44. A recent article6 used multiple regression to predict attitudes toward homosexuality. The researchers found thatthe effect ofnumber ofyears of education on a measure of tolerance toward homosexuality varied from essentially no effect for political conservatives to a considerably positive
11.43. The Economist magazine5 developed a qualityof-lifc index for nations as the predicted value obtained by regressing an average of lifesatisfaction scores from several surveys on gross domestic product (GDP, per capita, in dollars), life expectancy (in years), an index of political freedom
11.42. A 2002 study4 relating the percentage of a child's life spent in poverty to number of years of education completed by the mother and the percentage of a child's life spent in a single parent home reported the results shown in Table 11.20. Prepare a one-page report explaining how to interpret
11.41. A study3 of mortality rates found in the U.S. that states with higher income inequality tended to have higher mortality rates. The effect of income inequality disappeared after controlling for the percentage of a state's residents that had at least a high school education. Explain how these
11.40. For Example 11.2 on mental impairment. Table 11.19 shows the result of adding religious attendance as a predictor, measured as the approximate number oftimes the subject attends a religious service over the course of a year. Write a short report, interpreting the information from this table.
11.39. Analyze the "house selling price" data file at the text Web site (which were introduced in Example 9.10 on page 278), using selling price of home, size of home, number of bedrooms, and taxes. Prepare a short report summarizing your analyses and conclusions.
11.38. In about 200 words, explain to someone who has never studied statistics what multiple regression does and how it can be useful.
11.37. Table 9.13 on page 294 in Chapter 9 is the "UN data" data file at the text Web site. Construct a multiple regression model containing two explanatory variables that provide good predictions forthe fertility rate. How did you select this model? (Hint:One way is based on entries in the
11.36. For the previous exercise, repeat the analysis, excluding the observation for D.C. Describe the effect on the various analyses of this observation.
11.35. Using software with the "2005 statewide crime"data file at the text Web site, conduct a regression analysis of violent crime rate with predictors poverty rate, the percent living in urban areas, and percent of high school graduates. Prepare a report in which you state a research question you
11.34. Refer to the "OECD data" file at the text Web site, shown in Table 3.11 on page 62 of Chapter 3.Pose a research question about how at least two of the variables shown in that table relate to carbon dioxide emissions. Conduct appropriate analyses to address that question, and prepare a
11.33. Refer to the student data file you created in Exercise 1.12. For variables chosen by yourinstructor, fit a multiple regression model and conduct descriptive and inferential statistical analyses. Interpret and summarize your findings.
11.32. Repeat the previous exercise using y = college CPA with predictors high school GPA and number of weekly hours of physical exercise.
11.30. Exercise 11.11 showed a regression of violent crime rate on poverty rate and percent living in metropolitan areas. The estimated standardized regression coefficients are 0.473 for 11.31. Refer to the "Student survey" data file (Exercise 1.11 on page 8). Using software, conduct a regression
11.29. A multiple regression model describes the relationship among a collection of cities between y = murder rate (number of murders per 100,000 residents) and x\ = Number of police officers (per 100,000 residents)X2 = Median length of prison sentence given to convicted murderers (in years)xt, =
11.28. For a group of 100 children of ages varying from 3 to 15, the correlation between vocabulary score on an achievement test and height of child is 0.65.The correlation between vocabulary score and age for this sample is 0.85, and the correlation between height and age is 0.75.(a) Show that the
11.27. Refer to Examples 11.1 (page 322) and 11.8(page 347). Explain why the partial correlation between crime rate and high school graduation rate is so different from the bivariatc correlation. (This is an example of Simpson's paradox, which states that a bivariate association can have a
11.26. Refer to the previous exercise.(a) Find the partial correlation between y and x\, controlling for X2. Interpret the partial correlation and its square.(b) Find the estimate of the conditional standard deviation, and interpret its value.(c) Show how to find the estimated standardized
11.25. Table 11.18 shows results of regressing y = birth rate (BIRTHS, number of births per 1000 population) on xj = women's economic activity (ECON)and X2 = literacy rate (LITERACY), using UN data for 23 nations.(a) Report the value of each of the following:(i) ryXx (ii) ryX2 (iii) R 2(iv) TSS (v)
11.24. A multiple regression analysisinvestigatesthe relationship between y = college CPA and several explanatory variables, using a random sample of 195 students at Slippery Rock University. First, high school GPA and total SAT score arc entered into the model. The sum of squared errors is SSE =
11.23. Use software with the "house selling price" data file to allow interaction between number of bedrooms and number of bathrooms in their effects on selling price.(a) Interpret the fit by showing the prediction equation relating y and number of bedrooms for homes with (i) two bathrooms, (ii)
11.22. A study analyzes relationships among y =percentage vote for Democratic candidate, xi =percentage of registered voters who are Democrats,andx2 = percentageofregistered voters who vote in the election, for several congressional elections in 2006. The researchers expect interaction, since they
11.21. Exercise 11.11 showed a regression analysis for statewide data on y = violent crime rate, xi -poverty rate, and X2 = percent living in urban areas. When we add an interaction term, we get the prediction equation y = 158.9 — 14.72xi —1.29x2 + 0.76xiX2.(a) As the percentage living in urban
11.18. Refer to the previous exercise.(a) Find a 95% confidence interval for the change in the mean of y for a one-unit increase in 11.19. Use software with the "house selling price" data file atthe text Website to conduct a multiple regression analysis ofy = selling price of home (dollars), xi =
11.17. For a random sample of 66 state precincts, data are available on y = Percentage of adult residents who are registered to vote X| = Percentage of adult residents owning homes X2 = Percentage of adult residents who are nonwhitc X3 = Median family income (thousands of dollars)X4 = Median age of
11.16. Refer to Table 11.5 on page 330. Test Hq: 132-0 that mental impairment is independent of SES, controlling for life events. Report the test statistic, and report and interpret the P-valuc for (a) Ha:132 A 0, (b) Ha-. f32 < 0.
11.15. The GSS has asked subjects to rate various groups using the "feeling thermometer." The rating is between 0 and 100, more favorable as the score gets closer to 100 and less favorable as the score gets closer to 0. For a small data set from the GSS. Table 11.16 shows results of fitting the

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

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