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Statistical Methods For The Social Sciences 5th Edition Alan Agresti - Solutions

Using software, conduct the repeated-measures ANOVA of the anorexia data in Table 12.18 (page 373), available at the text website. Interpret results.
The General Social Survey asks respondents to rate various groups using the “feeling thermometer” on a scale of 0 (most unfavorable) to 100 (most favorable).We plan to study how the mean compares for rating liberals and rating conservatives, for ratings in 2016 and ratings in 1986. Explain why
Recently the General Social Survey asked respondents,“Compared with 10 years ago, would you say that American children today are (1) much better off, (2) better off, (3) about the same, (4) worse off, or (5) much worse off.” Table 12.33 shows opinion responses for 10 of the subjects on three
Refer to Table 12.15 (page 370) about the influence of three entertainment types on children.(a) Using software, conduct the repeated-measures analyses of Section 12.5.(b) Suppose you scored the influence categories(−3,−2, 0, 2, 3). What would this assume about the response categories?Repeat
For the 2014 GSS, Table 12.32 shows sample means of political ideology (higher values being more conservative), classified by gender and by race, for those over 50 in age. For H0: no interaction, software reports F = 21.7, df1 = 1 and df2 = 1081, and P-value
The prediction equation ˆy = 16+2s+3r+8(s×r)relates y = annual income (thousands of dollars), s = sex(s = 1 for men, s = 0 for women), and r = race (r = 1 for whites, r = 0 for blacks). By finding the four predicted means for this equation, show that the coefficient 8 of the interaction term is
The 26 students in a statistics class for social science majors at the University of Florida were surveyed about their attitudes toward divorce. Each received a response score according to how many from a list of seven possible reasons were regarded as legitimate for a woman to seek a divorce. The
Table 12.30 shows results of an ANOVA on y =depression index by gender and marital status (married, never married, divorced). State the sample size and fill in the blanks in the ANOVA table. Interpret results.
Table 12.29 summarizes responses on political ideology in the 2014 General Social Survey by religion and sex. The P-value is
In 2013, the U.S. Census Bureau reported that the population median income was $29,127 for white females,$26,006 for black females, $41,086 for white males, and$30,394 for black males.(a) Identify the response variable and the two factors, and show these medians in a two-way classification of the
In the United States, the Bureau of Labor Statistics recently reported that for males the current population mean hourly wage is $22 for white-collar jobs, $11 for service jobs, and $14 for blue-collar jobs. For females, the means are $15 for white-collar jobs, $8 for service jobs, and$10 for
For the 2014 GSS, when we regress y = number of hours per day watching TV on s = sex (1 = male, 0 = female)and religious affiliation (r1 =1 for Protestant, r2 = 1 for Catholic, r3 = 1 for Jewish, r1 = r2 = r3 = 0 for none or other), we get ˆy = 2.7 + 0.1s + 0.4r1 + 0.2r2 − 0.2r3.(a) Interpret
Using software with the Housesdata set at the text website, conduct an ANOVA for y = house selling price with factors whether the house is new and whether number of bathrooms exceeds two.(a) Using α = 0.05, test the hypothesis of no interaction between the factors in their effects on y.(b)
Table 12.13 on page 368 gave the prediction equationˆy = 5.23 − 1.77p1 − 1.24p2 − 0.01s relating political ideology to political party ID and to sex. Find the estimated means for the six cells, and show that they satisfy a lack of interaction.
A recent GSS asked, “What is the ideal number of kids for a family?” Table 12.28 shows results of evaluating the effects of gender and race.(a) Explain how to interpret the results of the F tests.(b) Let s = 1 for females and 0 for males, and let r = 1 for blacks and 0 for whites. The no
The sample means were 2.66 for white females, 2.62 for white males, 3.48 for black females, and 3.14 for black males. Explain how these results seem to be compatible with the results of the tests shown.
When we use the GSS to evaluate how the mean number of hours a day watching TV depends on sex and race, for subjects of age 18–25, we get the results shown in Table
A recent GSS asked, “Would you say that you are very happy, pretty happy, or not too happy?” and “About how many good friends do you have?” Table 12.26 summarizes results, with number of friends as the response variable.(a) State a research question you could answer with these data.(b)
For g groups with n = 100 each, we plan to compare all pairs of population means.We want the probability to equal at least 0.80 that the entire set of confidence intervals contains the true differences. For the Bonferroni method, which t-score multiple of the standard error should we use for each
In a study to compare customer satisfaction at service centers for PC technical support in San Jose(California), Toronto (Canada), and Bangalore (India), each center randomly sampled 100 people who called during a two-week period. Callers rated their satisfaction on a scale of 0 to 10, with higher
Refer to the previous exercise.(a) Suppose that the first observation in the second group was actually 9, not 1. Then, the standard deviations are the same, but the sample means are 6, 7, and 8 rather than 6, 3, and 8.Do you think the F test statistic would be larger, the same, or smaller? Explain
Table 12.24 shows scores on the first quiz (maximum score 10 points) in a beginning French course. Students in the course are grouped as follows:Group A: Never studied foreign language before, but have good English skills.Group B: Never studied foreign language before; have poor English
A recent GSS asked, “How often do you go to a bar or tavern?” Table 12.23 shows descriptive statistics and an ANOVA table for comparing the mean reported number of good friends at three levels of this variable.(a) State the (i) hypotheses, (ii) test statistic value,(iii) P-value, (iv) decision
Refer to the previous exercise. Table 12.22 shows an ANOVA table for the model.(a) Specify the hypotheses tested in this table.(b) Report the F test statistic value and the P-value. Interpret the P-value.(c) Based on (b), can you conclude that every pair of religious affiliations has different
A recent General Social Survey asked, “What is the ideal number of kids for a family?” Show how to define dummy variables, and formulate a model for this response with explanatory variable religious affiliation (Christian, Muslim, Jewish, Other or none).
A General Social Survey asked subjects how many good friends they have. Is this associated with the respondent’s astrological sign (the 12 symbols of the zodiac)?The ANOVA table for the GSS data reports F = 0.61 based on df1 = 11, df2 = 813.(a) Specify the null and alternative hypotheses for the
For GSS data comparing the reported number of good friends for those who are (married, widowed, divorced, separated, never married), an ANOVA table reports F = 0.80.(a) Specify the null and alternative hypotheses for the test.(b) Software reports a P-value of 0.53. Explain how to interpret it.(c)
Adjusted R2 is defined aswhere s2 is the estimated conditional variance and s2y is the sample variance of y, both of which are unbiased. This relates to ordinary R2 by(a) Suppose R2 = 0.339 for a model with p = 2 explanatory variables, as in Table 11.5.Find R2 adj when n = 10, 40 (as in the text
Let R2y(x1,... xp) denote R2 for the multiple regression model with p explanatory variables. Explain why Pyxxxp-1 R y(1p) 1-R2 R y(p-1) y(x1.Xp-1)
The numerator R2 – r2 yx1 of the squared partial correlation r2 yx2·x1 gives the increase in the proportion of explained variation from adding x2 to the model. This increment, denoted by r2 y(x2·x1), is called the squared semipartial correlation. One can use squared semipartial correlations to
Refer to the previous exercise. When we add an interaction term, we get ˆy = −16.6 + 66.6x1 − 31.8x2 +29.4(x1x2).(a) Interpret the fit by reporting the prediction equation between selling price and size of house separately for new homes (x2 = 1) and for old homes (x2 = 0). Interpret.(This fit
Chapters 12 and 13 show how to incorporate categorical explanatory variables in regression models. This exercise provides a preview. Table 11.25 shows some output for a model for the Houses2 data set at the text website, with y = selling price of home, x1 = size of home, and x2 = whether the house
Let ¯b∗i denote the estimated standardized regression coefficient when xi is treated as the response variable and y as an explanatory variable, controlling for the same set of other variables. Then, ¯b∗i need not equal b∗i. The squared partial correlation between y and xi, which is
Software reports four types of sums of squares in multiple regression models. The Type I sum of squares, sometimes called sequential SS, represents the variability explained by a variable, controlling for variables previously entered into the model. The Type III sum of squares, sometimes called
Suppose the correlation between y and x1 equals the multiple correlation between y and x1 and x2. What does this imply about the partial correlation ryx2·x1?Interpret.
Which of the following sets of correlations would you expect to yield the highest R2-value? Why?(a) ryx1= 0.4, ryx2= 0.4, rx1x2= 0.0.(b) ryx1= 0.4, ryx2= 0.4, rx1x2= 0.5.(c) ryx1= 0.4, ryx2= 0.4, rx1x2= 1.0.
Whenever x1 and x2 are uncorrelated, then R2 for the model E(y) = α+β1x1+β2x2 satisfies R2 = r2 yx1+r2 yx2 .In this case, draw a figure that portrays the variability in y, the part of that variability explained by each of x1 and x2, and the total variability explained by both of them together.
For the models E(y) = α + βx and E(y) =α + β1x1 + β2x2, express null hypotheses in terms of correlations that are equivalent to the following:(a) H0: β = 0.(b) H0: β1 = β2 = 0.
Give an example of three variables for which you expect β = 0 in the model E(y) = α + βx1 but for which it is plausible that β1 = 0 in the model E(y) = α + β1x1 + β2x2.
Let y = height, x1 = length of right leg, and x2 =length of left leg. Describe what you expect for the relative sizes of rx1x2, ryx2, R, and ryx2·x1.
Explain the difference in the purposes of the correlation, the multiple correlation, and the partial correlation.
The F test for comparing a complete model to a reduced model can be used to test(a) The significance of a single regression parameter in a multiple regression model.(b) H0: β1 = · · · = βp = 0 in a multiple regression equation.(c) H0: no interaction, in the model E(y) = α+β1x1 +β2x2 +β3x3
If ˆy = 2 + 3x1 + 5x2 − 8x3,(a) ryx3 < 0.(b) ryx3·x1 < 0.(c) ryx3·x1,x2 < 0.(d) Insufficient information to answer.(e) Answers (a), (b), and (c) are all correct.
If ˆy = 2 + 3x1 + 5x2 − 8x3,(a) The strongest correlation is between y and x3.(b) The variable with the strongest partial influence on y is x2.(c) The variable with the strongest partial influence on y is x3, but one cannot tell from this equation which pair has the strongest correlation.(d)
If ˆy = 2 + 3x1 + 5x2 − 8x3, then controlling for x2 and x3, the predicted mean change in y when x1 is increased from 10 to 20 equals(a) 3, (b) 30, (c) 0.3, (d) cannot be given—depends on specific values of x2 and x3.
In regression analysis, which of the following statements must be false? Why?(a) ryx1= 0.01, ryx2= −0.75, R = 0.2(b) The value of the residual sum of squares, SSE, can increase as we add additional variables to the model.(c) For the model E(y) = α + β1x1, y is significantly related to x1 at the
Table 11.24 shows results of fitting various regression models to data on y = college GPA, x1 = high school GPA, x2 = mathematics entrance exam score, and x3 =verbal entrance exam score. Indicate which of the following statements are false. Give a reason for your answer.(a) The correlation between
In Exercise 11.1 on y = college GPA, x1 = high school GPA, and x2 = college board score, E(y) = 0.20 + 0.50x1 + 0.002x2. True or false: Since β1 = 0.50 is larger than β2 = 0.002, this implies that x1 has the greater partial effect on y. Explain.
An article11 that analyzed the effects of the levels of the participant’s generosity and of the spouse’s generosity on a measure of marital quality reported that low levels of both were associated with low marital quality and high levels of both were associated with high marital quality.
An article10 used multiple regression to predict a measure of tolerance toward homosexuality.(a) The researchers found that the effect of number of years of education varied from essentially no effect for political conservatives to a considerably positive effect for political liberals. Explain how
The Economist magazine9 developed a quality-oflife index for nations as the predicted value obtained by regressing an average of life-satisfaction scores from several surveys on gross domestic product (GDP, per capita, in dollars), life expectancy (in years), an index of political freedom (from 1 =
A study8 relating the percentage of a child’s life spent in poverty to the 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 11.22.Prepare a one-page report explaining how to interpret the
A study7 of mortality rates found in the United States 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 who had at least a high school education.Explain how
For Example 11.2 on mental impairment, Table 11.21 shows the result of adding religious attendance as an explanatory variable, measured as the approximate number of times the subject attends a religious service over the course of a year. Write a report of about 200 words interpreting the table.
Analyze the Houses data file at the text website(and introduced in Example 9.10 on page 268), using selling price of home, size of home, number of bedrooms, and taxes. Prepare a one-page report summarizing your analyses and conclusions.
In about 200 words, explain to someone who has never studied statistics what multiple regression does and how it can be useful.
For the UN data file at the text website (Table 3.9 on page 53), construct a multiple regression model containing two explanatory variables that provide good predictions for the fertility rate. How did you select this model?(Hint: One way uses the correlation matrix.)
For the previous exercise, repeat the analysis, excluding the observation forD.C. Describe the effect of this observation on the various analyses.
Using software with the Crime data file at the text website, conduct a regression analysis of violent crime rate with the explanatory variables poverty rate, the percentage living in urban areas, and the percentage of high school graduates. Prepare a report in which you state a research question
For the OECD data file at the text website, shown in Table 3.13 (page 58), 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 one-page report summarizing your
Using industry-level data, a recent study6 analyzed labor’s share of income, measured as total compensation divided by total compensation plus the gross operating surplus. The authors predicted this would decrease as the degree of financialization of the company increased. Financialization was
Refer to the student data file created in Exercise 1.12. For variables chosen by your instructor, fit a multiple regression model and conduct descriptive and inferential statistical analyses. Interpret and summarize your findings.
Refer to the Students data file. Using software, conduct a regression analysis using either (a) y = political ideology with explanatory variables number of times per week of newspaper reading and religiosity, or (b) y =college GPA with explanatory variables high school GPA and number of weekly
A recent study5 analyzed the effect of x1 = work hours per day and x2 = commuting time to work on y = political participation. For the cluster sample of 1001 adult Americans, ¯x1 = 8.4 hours (s = 2.4) and ¯x2 = 19.8 minutes (s = 13.6). Political participation, which was a composite variable based
Amultiple regression model describes the relationship among a collection of cities between y = murder rate(number of murders per 100,000 residents) and x1 =number of police officers (per 100,000 residents), x2 = median length of prison sentence given to convicted murderers(in years), x3 = median
For the 2014 GSS, Table 11.20 shows estimates(with se values in parentheses) for four regression models for y = political party identification in the United States, scored from 1 = strong Democrat to 7 = strong Republican. The explanatory variables are number of years of education in model 1, also
Refer to Examples 11.1 (page 308) and 11.8(page 331). Explain why the partial correlation between crime rate and high school graduation rate is so different(including its sign) from the bivariate correlation.
Refer to the previous exercise.(a) Find the partial correlation between y and x1, controlling for x2. Interpret the partial correlation and its square.(b) Find the estimate of the conditional standard deviation, and interpret.(c) Show how to find the estimated standardized regression coefficient
Table 11.19 shows results of regressing y = birth rate (number of births per 1000 population) on x1 =women’s economic activity and x2 = literacy rate, using UN data for 23 nations.(a) Report the value of each of the following:(i) ryx1, (ii) ryx2, (iii) R2,(iv) TSS, (v) SSE, (vi) mean square
A multiple regression analysis investigates the relationship between y = college GPA and several explanatory variables, using a random sample of 195 students at Slippery Rock University. First, high school GPA and total SAT score are entered into the model. The sum of squared errors is SSE = 20.
Use software with the Houses 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 relatingˆy and number of bedrooms for homes with (i) two bathrooms, (ii) three bathrooms.(b)
A study analyzes relationships among y = percentage vote for Democratic candidate, x1 = percentage of registered voters who are Democrats, and x2 = percentage of registered voters who vote in the election, for several congressional elections in 2016. The researchers expect interaction, since they
Exercise 11.11 showed a regression analysis for statewide data on y = violent crime rate, x1 = poverty rate, and x2 = percentage living in urban areas. When we add an interaction term, we get ˆy = 158.9 − 14.72x1 −1.29x2 + 0.76x1x2.(a) As the percentage living in urban areas increases, does
Refer to the previous exercise.(a) Test the partial effect of number of bathrooms, and interpret.(b) Find the partial correlation between selling price and number of bathrooms, controlling for number of bedrooms.Compare it to the correlation, and interpret.(c) Find the estimated standardized
Use software with the Houses data file at the text website to conduct a multiple regression analysis of y =selling price of home (dollars), x1 = size of home (square feet), x2 = number of bedrooms, x3 = number of bathrooms.(a) Use scatterplots to display the effects of the explanatory variables on
Refer to the previous exercise. Find a 95% confidence interval for the change in the mean of y for a(a) 1-unit increase, (b) 50-unit increase in the percentage of adults owning homes, controlling for the other variables.Interpret.
For a random sample of 66 state precincts, data are available on y = percentage of adult residents who are registered to vote, x1 =percentage of adult residents owning homes, x2 = percentage of adult residents who are nonwhite, x3 = median family income (thousands of dollars), x4 =median age of
Refer to Table 11.5 on page 316. TestH0: β2 = 0 that mental impairment is independent of SES, controlling for life events. Report the test statistic, and report and interpret the P-value for (a) Ha: β2 = 0, (b) Ha: β2 < 0.
The General Social Survey 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 11.17 shows results of
Table 11.16 comes from a regression analysis4 of y = number of children in family, x1 = mother’s educational level in years (MEDUC), and x2 = father’s socioeconomic status (FSES), for a random sample of 49 college students at Texas A&M University.(a) Write the prediction equation. Interpret
For 2014 GSS data on y = highest year of school completed, x1 = mother’s highest year of school completed, and x2 =father’s highest year of school completed, we obtain ˆy = 9.86 + 0.345x1 (r2 = 0.195), ˆy = 10.15 +0.330x2 (r2 = 0.204), and ˆy = 9.30 + 0.194x1 + 0.212x2(R2 = 0.243). In a
Refer to the previous exercise.(a) Report the F statistic for testing H0: β1 = β2 = 0, report its df values and P-value, and interpret.(b) Show how to construct the t statistic for testing H0:β1 = 0, report its df and P-value for Ha: β1 = 0, and interpret.(c) When we add x3 = percentage of
Table 11.14 shows Stata output from fitting the multiple regression model to recent statewide data, excluding D.C., on y = violent crime rate (per 100,000 people), x1 = poverty rate (percentage with income below the poverty level), and x2 = percentage living in urban areas.(a) Report the prediction
For recentUNdata for several nations, a regression of carbon dioxide use (CO2, a measure of air pollution) on gross domestic product (GDP) has a correlation of 0.786.With life expectancy as a second explanatory variable, the multiple correlation is 0.787.(a) Explain how to interpret the multiple
Recent UN data from several nations on y = crude birth rate (number of births per 1000 population size), x1 = women’s economic activity (female labor force as percentage of male), and x2 = GNP (per capita, in thousands of dollars) has prediction equation ˆy = 34.53 −0.13x1 − 0.64x2. The
Refer to the previous exercise. Using software with the Florida data file at the text website,(a) Construct box plots for each variable and scatterplots and partial regression plots between y and each of x1 and x2. Interpret these plots.(b) Find the prediction equations for the (i) bivariate
The Florida data file, shown partly on page 283, has data from the 67 Florida counties on y = crime rate(number per 1000 residents), x1 = median income (thousands of dollars), and x2 = percentage in urban environment.(a) Figure 11.12 shows a scatterplot relating y to x1. Predict the sign that the
Refer to the previous exercise.(a) Show how to obtain R-squared from the sums of squares in the ANOVA table. Interpret it.(b) r2 = 0.78 when GDP is the sole predictor. Why do you think R2 does not increase much when cell phone use is added to the model, even though it is itself highly associated
A regression analysis with recent UN data from several nations on y = percentage of people who use the Internet, x1 = per capita gross domestic product (in thousands of dollars), and x2 = percentage of people using cell phones has results shown in Table 11.12.(a) Write the prediction equation.(b)
Use software with the Crime2 data file at the text website, with murder rate (number of murders per 100,000 people) as the response variable and with percentage of high school graduates and the poverty rate as explanatory variables.(a) Construct the partial regression plots. Interpret. Do you see
The Social Progress Index (see www.socialprogressimperative.org) is a measure of national progress in delivering social and environmental value.It is an average of three component measures: BHN =basic human needs, incorporating basic medical care and personal safety; FW = foundations of well-being,
For recent data in Jacksonville, Florida, on y = selling price of home (in dollars), x1 =size of home (in square feet), and x2 = lot size (in square feet), the prediction equation is ˆy = −10,536 + 53.8x1 + 2.84x2.(a) A particular home of 1240 square feet on a lot of 18,000 square feet sold for
For students at Walden University, the relationship between y = college GPA (with range 0–4.0) and x1 =high school GPA (range 0–4.0) and x2 = verbal college board score (range 200–800) satisfies E(y) = 0.20 +0.50x1 + 0.002x2.(a) Find the mean college GPA for students having (i)high school GPA
4.44.* Refer to the previous exercise. Describe the sampling distribution of ¯y(a) for a random sample of size n = 1;(b) if you sample all 50,000 residents.
9.65.*Refer to the previous exercise. Let ρ1 and ρ2 denote the population correlation values between two variables for two separate populations. Let r1 and r2 denote sample values for independent random samples from the populations.To test H0: ρ1 = ρ2, the test statistic iswhere T1 and T2 are
9.67.* The formula for the correlation can be expressed as(a) Using the first formula, explain why the correlation has the same value when x predicts y as when y predicts x.(b) By the second formula, the correlation is approximately the average product of the z-score for x times the z-score for y.
9.69.* Suppose that the linear regression model E(y) = α + βx with normality and constant standard deviation σ is truly appropriate. Then, the interval of numberspredicts where a new observation on y will fall at that value of x. This interval, which for large n is roughly ˆy ± 2s, is a 95%
In murder trials7 in 20 Florida counties in two years, 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 killed a
Consider the relationship between y = political party preference (Democrat, Republican) and x1 = race(Black, White) and x2 = gender. There is an association between y and both x1 and x2, with the Democrat preference being more likely for blacks than whites and for women than men.(a) x1 and x2 are
Example 9.10 (page 265) used a data set on house sales to regress y = selling price of home (in dollars) to x= size of house (in square feet). The prediction equation was ˆy = −50,926 + 126.6x. Now, we regard size of house as x1 and also consider x2 = whether the house is new (yes or no). The

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