Statistical Methods for the Social Sciences

Alan Agresti, Barbara Finlay

Chapter 12 Comparing Groups: Analysis of Variance Methods - all with Video Answers

Educators

Chapter Questions

Problem 1

Studics of the degree of residential scgregation betueen blacks and w hites use the segre. gation index, defincd as the percentage of nonwhites who would have to change the block on which thcy live in order to produce a fully nonsegregated city-one in which the percentage of nonwlutes living in each block is the same for all blocks in the city This index can assume values ranging from 0 to 100 , with higher values indicating greater segregation. (The national average was 65 in 1990.) Table 12.25 shows the index for a sample of cities in 1994, classified by region. In no sense is this a random sample. but use it to illustuate mechanics of one-way ANOVA. The ovcrall mean is 72.75 , and $\bar{Y}_1=81, \bar{Y}_2=88$, $\bar{Y}_7=71 . \bar{Y}_4=51$, with $s_1=4.69, s_2=2.58, s_3=4.83, s_4=12.46$. Consider the null hypothesis $H_0: \mu_1=\mu_2=\mu_3=\mu_4$, where $\mu_i$ is the mean for all cities in region $i$.

Maxime Rossetti

Numerade Educator

Problem 2

Refer to the previous exercise.
a) State the assumptions for the analysis, and show that at least two of them may be riolated. What is the impact?

Check back soon!

12:20

Problem 3

A consumer protection group compares three diffel cut types of front bumpers for a bi and of automobile A test is conducted by driving an dutomobile into a brick wall at 15 mules per hour. The response is the amount of damage to the car, as measured by the repair costs, in hundreds of dollars Due to the potentially largc costs. the study conducts only two tests with each bumper type. Table 12.26 shows the 1 csults

Diane Koenig

Numerade Educator

11:57

Problem 4

Table 1227 show's scorcs on the first quit (maximum scorc 10 points) in a beginning French course. Students in the course are grouped as follous:
Group A. Never studied forejgn language before, but have good English skills
Group B: Never studied Ioreign language before have poor English skulls
Group C Sudied other foreign language
a) Treating the students in this course as a random sample of all ninth-grade studenls taking this beginning course, show all steps of a statistical test of whether the three population means differ on this quiz. Use Table 12.28 , based on using SPSS to perform the analyses. to provide results of calculations Interpret the result
b) Use sinnukaneous $85 \%$ confidence inter als to compare pais of means Interpret each interval Construct a diagram to indicate the means. il any that arc significantly different

Jeremiah Mbaria

Numerade Educator

Problem 5

The General Social Survey dshs respondents to rate various gtoups using the "feeling thermometer" Ratings betwreen 50 and 100 mean you feel lavorahle and warm toward the group, whereas ratings between 0 and 50 mean that you don't feel fas orable. When asked to rate liberals, the mean response was 566 during 1983-87 ( $n=1351, s=20.7$ ) and 578 for an independent sample during 1988-91 ( $n=2228.5=25.0$ ). Repont the ANOVA $F$ statistic for comparing the mean responses (You can either construct $F$ or use the connection $F=t^2$ for the lest stalistic from the two-sample $f$ test.)

Check back soon!

Problem 6

Refer to Table 12.19.
a) Using software, conduct a one-way A.VOVA for the 72 observations at time $=$ after. Verify that the $F$ test stacistic companng the three treatmenis at that time equals 865 , based on $d f_1=2$ and $d f_2=69$, for which $P=.0004$.
b) Show how to obtain the sane results by conducting a regression analysis.

Check back soon!

Problem 7

Table 1229 is a contingency table sumnaarizung responses on political ideology in the 1991 General Socjal Survey by race and gender. Table 12.30 shows results of using SAS software to conduct an analysis of vanance companng the four groups, using the category labels for the ideology scores.
a) Show all steps of a one-way ANOVA to test the hypothesis of equal population mcans. Report the $P$-value and interpret.
b) The printout does not show the actual Bonferroni confidence intervals, but it shous the $f$-score used for simultaneous $95 \%$ intervals. Show how to construct the one companng the fist tho groupss

Check back soon!

Problem 8

Refer to the previous exercise, using the black sample only
a) Conduct an ANOVA comparing the two gioups. Interpret.
b) How does your analysis relate io the methods of Chapter 7 for comparing means?
c) How, if at all. would the results differ if you used ideolog) scores (i) $(-3,-2,-1,0$. $1,2,3)$. (u) $(10.20,30,40,50.60 .70)^?$

Check back soon!

Problem 9

Use software with the data in Table 9.4
a) Conduct au ANOV'A to test equality of the mean selling prices for homes with one. two, and three bathrooms. Interpret.
b) Explain the difference between conducting a test of independence of selling price and number of bathroonis using the ANOVA test in (a) and using a regression $t$ test for the coefficient of the number of bathrooms in a regressiou model.
c) Show how to conduct a regression analysis that is equivalent to the ANOVA in (a).
d) Use a muluple companson method to construcl simultaneous $85 \%$ confidence intervals comparing the means. Interpret. How do these inter als comparc to ordiuary $95 \%$ confidence intervals for individual comparisons?

Check back soon!

13:31

Problem 10

. Refer to lable 9.4 In that data set, whether a house is new is a dummy vanable Using soltwarc, put this as the sole predictor of sclling price in a regression analysis.
a) Conduct the $t$ test for the effect of this variable in the regression analysis (i.e., test $H_0$ : $\beta=0$ ). Interpret
b) Conduct the $F$ test for the analysis of vanance comparing the mean selling prices of new and exısting homes.
c) Explain the conncction between the value of $t$ in (a) and the value of $F$ in (b).

Oluwadamilola Ameobi

Numerade Educator

01:37

Problem 11

A psychologist compaues the mean amount of time of REM sleep for subjects under three conditions. She uses three groups of subjects, with four subjects in each group. Table 12.31 shows a S.AS printout for the analysis.
a) Report and iuterpret all steps of the ANOVA $F$ test.
b) Explain how to obtain the plus and minus part of 13.96 for the $95 \%$ Bouferroni confidence intervals.
c) Set up dummy vanables for a regression model so that an $F$ test for the regression pdrameters is equivalent to the ANOVA test. Express the null hypothesis both in terms of population means and regression parameters
d) Report the prediction equation obtained in firting this regression equation

Dominador Tan

Numerade Educator

01:11

Problem 12

For $g$ groups with large sample sizes, we plan to compare sinrultaneously all pairs of population means. We want the probability to equal at least 80 that the enture set of confidence intervals contain the true diffesences.
a) If $g=10$, which tahled $t$-score should we use lor each inter al?
b) Which $t$-score should we use if $g=5^{\circ}$
c) Descnbe how the $t$ - (or 2 -) score depends on the number of groups. and explain the implication regarding width of the intervals.

Maxime Rossetti

Numerade Educator

Problem 13

A geographer compares residential lot sizes in four quadrants of a cily To do this, he randonly samples 300 records from a city file on home residences and records the lot sizes (in thousands of square feet) by quadrant. The ANOVA rable (Table 12.32) relers to a comparison of mean lot sizes for the northeas! (NE), northwest (NW). soutliwest (SW). and southeast (SE) quadrants of the city.

Check back soon!

Problem 14

A study comnpares the mean level of contnbutions to political campaigns in Pennsyh ania by regisiered Democrats, regstered Republicans, and unaffiliated voters
a) Write a regression equation for this analysis, and interpret the parameters in the oded.
b) Explain hew one can use the regression model to test the null hypothesis of equal mean contributions for the three groups.

Check back soon!

Problem 15

Use computer sof(ware to reproduce the one-u ay ANOVA results reported in Table 12.2 for Example 121 .
1

Check back soon!

13:15

Problem 16

According to the U.S Department of Labor, the mean hourly wage in 1994 was $\$ 15.71$ fol a college graduate and $\$ 9.92$ for a high school graduale. In 1979, the means (in 1994 dollars) were $\$ 15.52$ for college graduates and $\$ 11.23$ for high sclool graduates.
a) Construct a $2 \times 2$ table to display the results.
b) Comparc the differences between 1994 and 1979 (i) for college graduates. (ii) for high school graduates. Is there interaction? Explain.
c) Show a set of four means that would display an absence of interaction.

Lyle Anderson

Numerade Educator

View

Problem 17

Refer to Table 7.18 in Problem 7.40, which summanzes the mean number of dates in the past three months by gender and hy level of physical atuactiveness. Do these data appear to show interaction? Explain.

Rashmi Sinha

Numerade Educator

Problem 18

A recent regression analysis of college faculty salaries in 1984 (M. Bellas, American Sociological Rericw: Vol. 59, 1994, p. 807) included a large number of predictors, including a dummy variable for gender (male $=1$ ) and a dummy variable for race (nonwhite $=1$ ) For annual incoine measured in thousands of dollars. the estinuated coeflicients were . 76 for gender and .62 for race.
a) Interpret these coefficients
b) At particular settings of the other predictors, the cstimated mean salary for while females was 30.2 thousand. Find the estimated means for the other three groups.
c) Show hou to use the coefficients 10 find the estimated difference in mean income between nonwhite male facuity and white female faculty.

Check back soon!

02:45

Problem 19

Refer to the predicuon equation $\hat{Y}=458-.71 P_1-.54 P_2-.08 G$ for the no interaction model in Example 12.5.
a) Using it. find the estimated means for each of the six cells. and show that they satisfy a lack of interaction.
b) Using results for this model, consuuct and interpret $95 \%$ Bonfcrrom confidcnce intervals companing pairs of party IDs Compare results to those in Table 12.3 for the one-way ANOVA. Which pars are significautly different?

Khoobchandra Agrawal

Numerade Educator

Problem 20

Use software with Table 12.11, analyzed in Section 12.5.
a) Fit the no interaction model, and venty the resulls given there.
b) Fil the interaction model. Compare SSE for this model to SSE for the no intelaction model. Show how the difference betu cen these values relates to the partial sum of squares for testing interaction.
c) Using the prediction equation for the interaction model. find the six estimated cell means, and compare them to the sample means (Note- The model uses six parameters to summarize six means, so it has a perfect fit)

Check back soon!

Problem 21

Consider the regression model $E(Y)=\alpha+\beta_1 G+\beta_2 R+\beta_3(G R)$, where $Y=$ income (thousands of dollars), $G=$ gender ( $G=1$ for men and $G=0$ for women), and $R=$ race ( $R=1$ for whites and $R=0$ for blacks).

Check back soon!

Problem 22

. Rcfer to Problem 12.7. Tabie 12.33 shows the result of using SAS to conduct a two-way ANOVA. with both gender and race as predictors. The first panel of the table shows the result of the model with an interaction term. and the other three pancls show results after dropping that term.
a) Test the hypothesis of no interaction. Report the $F$ test statistic, the $P$-value, and inlerpret.
b) Test the hy pothesis for the rave main eflect, controlling for gender. Report the $F$ test statistic, the $P$-value. and interpret.
c) Test the hypothesis for the gender main effect, controlling for race Report the $F$ test statisuc, the $P$-yaluc, and interpret
d) Construct a $95 \%$ coufidence interval for the gender main effect. and interpret.
e) Defining dummy variables. explain how the tests in (a)-(c) ,elate to regression models.

Check back soon!

Problem 23

Tablc 12.34 show s results of an ANOVA on $Y=$ deplession index and the predictors gender and nuarital status (married, never manied, divorced).
a) State the regression model for this analysis.
b) State the sample size and fill in the blunks in the ANOVA table
c) Interpret the result of the $F$ test for no interaction

Check back soon!

01:27

Problem 24

The 25 women faculty in the humanities division of a college have a mean salary of $\$ 46,000$, whereas the five in the science div sision have a mean salary of $\$ 60,000$.

Sheryl Ezze

Numerade Educator

Problem 25

Refor to Example 12.6 and Table 12.16.
a) Using sotiware, conduct the repeated meisures analyses of Section 12.6
b) Suppose one scored the influcnce categones (1, 2.3.4.5). Would this have any effect on the test statistic Explain
c) Suppose one used scores ( $-3,-20,2,3)$. What would this dssume about the response categolies? Repeat the analyses using these scores Are the conclusions sensitive to the choice of scoles?

Check back soon!

View

Problem 26

Recently the General Social Survey acked respondents. "Compared with ten years ago. would you say that Amelican childien today arc (1) much betteı oft, (2) better off, (3) about the same. (4] worse off. on (5) much worse off "Table 12.35 shows iesponses for ten of the subjects on three issucs: quality of their education. safety of the neighborhoods they live in. and getting health care when they need it.

Lainey Roebuck

Numerade Educator

01:43

Problem 27

Refer to the previous exercise. The first five respondents were iemale, and the last five were male. Analyze these data using both gender and issue as factors.
a) Identify the between-subjects and within-subjects factors.
b) Test for interaction. Interpret.
c) Test the main effect of gender. Interpret.
d) Test the main effect of issue Interpret.
c) Use $95 \%$ confidence intervals to compare the means for the three issucs Interpret, and summarize your findings from these analyses

James Kiss

Numerade Educator

Problem 28

Refer to Problem 125 When these subjects werc asked to rate conser atives. the mean resjonses were 61.9 during 1983-87 and 60.7 during 1988-91. Explain why a two-way ANOV'A using time (1983-87. 1988-91) and group rated (liberal, Conservative) as factors would require methods for repeated measures Identify the withm-subjects and between-subjects factors. Would you expect the test of no interaction to yield a sinall $P$-valuc? Explain.

Check back soon!

Problem 29

Upjohn, a pharmaceutical company, conducted a randomized clinical tnal comparing an active hypnotic drug with a placebo for patients suffering from insomnia. The outcome is patient response to the question, "How quickly did you fall asleep after going to bed?" Patients sufferng from insomnia were randomly assigned to receive the active drug or a placebo. The study measured patients' responses at the start and at the conclusion of a two-week treatment period.

Check back soon!

Problem 30

Table 12.37 shows results of using SPSS in Example 12.7 with Table 1219
a) Explain hou to use the infornation in this table to conduct the test of no interaction between treatment and time.
b) Explain how to deterninc the df values for tieatment. time. and their inteaction.

Check back soon!

Problem 31

Using softw are, conduct the Icpoated measules ANOVA in Example 12.7 with Table 12.19 .

Check back soon!

Problem 32

. Refer to the WWW data sel (Problem 1.7). with 1esjonse variable the number of weckly hours engaged in sports and other physical exercise. Using software, conduct an analysis of variance and iollow-up estimation, and prepare a 1 cport summarizing you analyses and interpretations using
a) Gender as a predictor.
b) Gender and $u$ hether a segelarian as predictors

Check back soon!

Problem 33

Refer to the data hle created in Problens 1.7 For variables chosen by you instructor. usc ANOVA methods and related inferential statistical analyses. Inlerpret and summarize youn findings.

Check back soon!

00:41

Problem 34

The General Secial Surve) has frequently asked respondents how they would rate vanous countries on a scale from -5 to +5 , where -5 indicates the country is disliked very much and +5 indicates the country is liked ver much. Table 12.38 shows counts of the responses regarding ratings of Russia for surveys from various years. Using software, analyzc these data. Preparc a rcport providing jour analyses and showing interpretations and conclusions.

Brandon Fox

Numerade Educator

Problem 35

A random sample of 26 female students at a major university were surneyed abous their attitudes Io and abortion. Each received a score on abortion attitudc according to bow many from a list of eight possible reasons for abortion she would accept as a legitimate reason for a woman to seek abortion Thus. the higher the score. the mole favor able the attitude toward abortion as an option in a vat iety of circunstances. T he students were classified as "Iundamentalist" or "monfundamentalist" in their religious beliefs. They were also classified according to theu chunch attendance frequency, "frequent" (more than oncc a month) or "infrequent " Table 12.39 displays the 26 ahortion attitude scores, classified by religion and frequency of church antendance. Using software, analyze the data Discuss your findings in a short report, indicating the models ftted, hypotheses tesied. pasameters estimated, and interpretations that follow from your analy ses.

Check back soon!

00:19

Problem 36

Table 12.40 , based on data from the 1989 General Social Survey, is a contingency table summarizing responses of 29 subjects regarding government spending on the environ

Nick Johnson

Numerade Educator

03:22

Problem 37

Refer to the WWW data set in Problcm 1.7. Use 1epcatcd measures analyses to model the weekly number of hours of recreation in terms of type of activity (levels $S=$ sports and physical exercise, $T=\mathrm{TV}$ watching) and gender Interpret results of all analyses. and suminanze.

Sheryl Ezze

Numerade Educator

View

Problem 38

An experiment uscd four randomly selected groups of five individuals each. The overall sample mean was 60 .
a) What did the data look like if the one-way ANOVA for compangg the means had test statistic $F=0$ ?
b) What did the data look like if $F=\infty$ ?

Victor Salazar

Numerade Educator

01:35

Problem 39

Explain carefully the difference between a probability of Type I error of .05 for a single companson of two means aud a multiple comparison error rate of 05 for comparing all pairs of means

Narayan Hari

Numerade Educator

Problem 40

In multiple compansons following a one-way ANOVA, is it possible that group A is not significantly different Irom group B and group B is not significantly different Irom group C yet group A is significantl) differ ent from group C? Explain.

Check back soon!

01:33

Problem 41

Construct a numerical exanple of means for a two-way classification under the following conditions:
a) Main elfects ale present only for the row varable.
b) Main effects are present for each variable, but there is no interaction.
c) Interaction effects are present.
d) No effects of any type are piesent.

Dominador Tan

Numerade Educator

02:40

Problem 42

The null hypothesis of cquaslity of means for a factor is rejected in a (wo-way ANOVA Does this unply that the hypothesis will be rejected in a one-way ANOVA $F$ test. is the data are collapsed over the levels of the second vanable? Explain.

Erin Moser

Numerade Educator

01:33

Problem 43

For a two-way classification of means by factors $A$ and $B$, at each level of $B$ the means are equal fol the levels of $A$. Docs this imply that the overall means are equal at the various levels of A. ignoring $B^7$ Explain the implications, in terms of how results may differ between Iwo-uay ANOVA and one-way ANOVA.
Select the correct response(s) in Problemis 12.44-12.47.

Dominador Tan

Numerade Educator

Problem 44

Analysis of vartance and regression are similar in the sence that
a) They both assume a quantitative response variable
b) They both have $F$ tests for testing that the responsc variable is statistically independent of the explanatory variable(s)
c) For inferential purposes, they both assume that the response variable $Y$ ' is normally distibuted with the same standard deviation at all combinations of levels of the explanatory variable(s).
d) They both provide ways of partitioning the sariation in $Y$ into "cxplained" and "unexplained" components.

Check back soon!

Problem 45

Onc-way ANOVA provides relatively more evidence that $H_0: \mu_1=\cdots=\mu_s$ is false
a) The smaller the "between" vanation and the larger the "within" variation.
b) The smaller the "berween" variation and the smaller the "within" variation.
c) The largel the "between" variation and the smaller the "within" variation.
d) The larger the "berween" variation and the larger the "within" variation.

Check back soon!

02:05

Problem 46

For four means. a multiple conıparison method provides sinultaneous $95 \%$ confidence intervals for the differcnces between the six pairs. Then
a) For each confidence inter al. there is a .95 chance that it contans the truc difference.
b) The probability that all six confidence intervals ane correct is .70 .
c) The probability that all six confidence intervals are correct is .95
d) The probability that all six confidence intervals are correct is $(.95)^6$.
c) The probability is .05 that at least one cunfidence interval does not contan the trac difference.
f) The confidence intervals are wider than separate $95 \%$ confidence intervals for each difference.

Sheryl Ezze

Numerade Educator

Problem 47

Interaction tenns are needed in a two-way ANOVA model when
a) Each pain of vanables is associated.
b) Bolh explanatory variables have significant effects in the model without interaction terms
c) The difference in means between two categories of one explanarory variable varics greatly among the categones of the other explanatory variable.
d) The mean squarc for interaction is huge compaed to the mcan square error.

Check back soon!

02:51

Problem 48

You know the sample mean. standard deviation. and sample size for each of three groups. Can you conduct an ANOVA $F$ test. or would you need more information?

Beth Stone

Numerade Educator

02:05

Problem 49

You form a $95 \%$ conlidence interval in five different situations.
a) Assumng that the results of the intervals are statistically independent. find the probabilty that all five intervals contain the parameters they are designed to estimate. Find the probability that at least one interval is in error. (Hint Use the binomial distnbution)
b) What conhdence coefficient should you use for each interval so that the probability that all five intervals contain the paaameters equals exactly .95 ? Compare this to the confidence cocfficient for each interval used in the Bonferroni method.

Sheryl Ezze

Numerade Educator

View

Problem 50

Show how to construct a regression model for the analy'sis of mean income over a threeway classIfication of gender (male, female). race (white, black), and type of job (assemblyline worker. janitorial). Interpret the parameters in the model. Assume that there is no interaction

Shu Naito

Numerade Educator

View

Problem 51

This exercisc motivates the formula for the berween estimate of the variance in one-way ANOVA. Suppose the sample sizes all equal $n$ and the population means all equal $\mu$. The samplung distribution of each $\bar{Y}$, then has mean $\mu$ and vanance $\sigma^2 / n$. The sample mean of the $\bar{Y}_i$ talues, when the sample sizes are equal. is simply $\bar{Y}$.
a) Trcatmg $\bar{Y}_1, \bar{Y}_2, \ldots, \bar{Y}_g$ as $g$ observations having the sample mean $\bar{Y}$, explain why $\sum\left(\bar{Y}_i-\bar{Y}\right)^2 /(g-1)$ estimates the variance $\sigma^2 / n$ of the distribution of the $\bar{Y}_i$-values. b) Using (a). explain why $\sum n\left(\bar{Y}_i-\bar{Y}\right)^2 /(g-1)$ estimates $\sigma^2$. For the unequal sample sizc case, replacing $n$ by $n$, yields the between estimate.

Victor Salazar

Numerade Educator

Problem 52

The Scheffe multiple comparison method for comparing $g$ groups with overall error rate $\alpha$ provides interval for $\mu_i-\mu_j$ of
$$ \left(\bar{Y}_1-\bar{Y}_j\right) \pm \hat{\sigma} \sqrt{(g-1) F_Q\left(\frac{1}{n_i}+\frac{1}{n_j}\right)} $$
where $F_q$ denotes the value fiom the $F$ distribution with $d f_1=g-1$ and $d f_2=N-g$ having right-tail probability $\alpha$. Apply this method to Example 12.3. interpret results. and comparc the intervals to the ones for the Bonferroui method

Check back soon!