Statistics: A Tool For Social Research

Joseph F. Healey

Chapter 11 Hypothesis Testing IV: Chi Square - all with Video Answers

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Chapter Questions

Problem 1

For each of the following tables, calculate the obtained chi square. (HINT: Calculate the expected frequencies for each cell with Formula $11.2 .$ Double-check to make sure you are using the correct row and column marginals for each cell. It may be helpful to record the expected frequencies in table format as well-see Tables $11.2,11.4$, and $11.7 .$ Next, use a computational table to organize the calculation for Formula I1.l-see Tables $11.5$ and 11.8. For each cell, subtract expected frequency from observed frequency and record the result in column 3. Square the value in column 3 and record the result in column 4, and then divide the value in column 4 by the expected frequency for that cell and record the result in column 5. Remember that the sum of column 5 in the computational table is the

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Problem 2

Many analysts have noted a "gender gap" in elections for the U.S. presidency, with women more likely to vote for the Democratic candidate. A sample of university faculty has been asked about their a. Is there a statistically significant relationship between gender and party preference?
b. Compute column percents for the table to determine the pattern of the relationship. Which gender is more likely to prefer the Democrats?

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Problem 3

A local politician is concerned that a program for the homeless in her city is discriminating against blacks and other minorities. The following data were taken from a random sample of black and white homeless people.
\begin{tabular}{lccc}
& \multicolumn{2}{c} { Race } & \\
\cline { 2 - 3 } Received Services? & Black & White & Totals \\
\hline Yes & 6 & 7 & 13 \\
No & $\underline{4}$ & $\frac{9}{16}$ & $\frac{13}{26}$
\end{tabular}
a. Is there a statistically significant relationship between race and whether or not the person has received services from the program?
b. Compute column percents for the table to determine the pattern of the relationship. Which group was more likely to get services?

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Problem 4

A sample of 25 cities have been classified as high or low on their homicide rates and on the number of handguns sold within the city limits. Is there a relationship between these two variables? Do cities with higher homicide rates have significantly higher gun sales? Explain your results in a sentence or two.
\begin{tabular}{lccc}
& $\underline{\text { Homicide Rate }}$ & \\
Volume of & & & \\
Gun Sales & Low & High & Totals \\
\hline High & 8 & 5 & 13 \\
Low & $\frac{4}{12}$ & $\frac{8}{13}$ & $\frac{12}{25}$
\end{tabular}

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Problem 5

Is there a relationship between salary levels and unionization for public employees? The following data represent this relationship for fire departments in a random sample of 100 cities of roughly the same size. Salary data have been dichotomized at the median. Summarize your findings.
\begin{tabular}{lccc}
& \multicolumn{2}{c} { Status } & \\
\cline { 1 - 3 } Salary & Union & Nonunion & Totals \\
\hline High & 21 & 29 & 50 \\
Low & $\frac{14}{35}$ & $\frac{36}{65}$ & $\frac{50}{100}$
\end{tabular}
a. Is there a statistically significant relationship between these variables?
b. Compute column percents for the table to determine the pattern of the relationship. Which group was more likely to get high salaries?

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Problem 6

A local judge has been allowing some individuals convicted of "driving under the influence" to work in a hospital emergency room as an alternative to fines, suspensions, and other penalties. A random sample of offenders has been drawn. Do participants in this program have lower rates of recidivism for this offense?
stat \begin{tabular}{lccc}
& \multicolumn{2}{c} { Non- } \\
Recidivist? & Participants & participants & Totals \\
\hline Yes & 60 & 123 & 183 \\
No & 55 & $\frac{108}{231}$ & $\frac{163}{346}$
\end{tabular}
a. Is there a statistically significant relationship between these variables?
b. Compute column percents for the table to determine the pattern of the relationship. Which group is more likely to be rearrested for driving under the influence?

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Problem 7

The state Department of Education has rated a sample of local school systems for compliance with state-mandated guidelines for quality. Is the quality of a school system significantly related to the affluence of the community as measured by per capita income?

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Problem 8

SOC A program of pet therapy has been running at a local nursing home. Are the participants in the program more alert and responsive than nonparticipants? The results, drawn from a random sample of residents, are reported here.
\begin{tabular}{lccc}
& \multicolumn{3}{c} { Status } \\
\cline { 3 } Alertness & Participants & \multicolumn{1}{c} { Non- } \\
participants & Totals \\
\hline High & 23 & 15 & 38 \\
Low & $\frac{11}{34}$ & $\frac{18}{33}$ & $\frac{29}{67}$
\end{tabular}
a. Is there a statistically significant relationship between participation and alertness?
b. ompute column percents for the table to determine the pattern of the relationship. Which group was more likely to be alert?

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Problem 9

SOC Is there a relationship between length of marriage and satisfaction with marriage? The necessary information has been collected from a random sample of 100 respondents drawn from a local community. Write a sentence or two explaining your decision.
a. Is there a statistically significant relationship between these variables?
b. Compute column percents for the table to determine the pattern of the relationship. Which group is more likely to be highly satisfied?

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Problem 10

Is there a relationship between college class standing and political orientation? Are juniors and seniors significantly different from freshmen and sophomores on this variable? The following table reports the relationship between these two variables for a random sample of 267 college students. a. Is there a statistically significant relationship between these variables?
b. Compute column percents for the table to determine the pattern of the relationship. Which group is more likely to be conservative?

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Problem 11

At a large urban college, about half of the students live off campus in various arrangements, and the other half live in dormitories on campus. Is academic performance dependent on living arrangements? The results based on a random sample of 300 students are presented here. a. Is there a statistically significant relationship between these variables?
b. Compute column percents for the table to determine the pattern of the relationship. Which group is more likely to have a high GPA?

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Problem 12

An urban planning commissioner has built up a database describing a sample of the neighborhoods in her city and has developed a scale by which each area can be rated for the "quality of life" (this includes measures of pollution, noise, open space, services available, and so on). She has also asked samples of residents of these areas about their level of satisfaction with their neighborhoods. Is there significant agreement between the commissioner's objective ratings of quality and the respondents' self-reports of satisfaction? a. Is there a statistically significant relationship between these variables?
b. Compute column percents for the table to determine the pattern of the relationship? Which group is most likely to say that their satisfaction is high?

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Problem 13

Does support for the legalization of marijuana vary by region of the country? The table displays the relationship between the two variables for a random sample of 1020 adult citizens. Is the relationship significant? a. Is there a statistically significant relationship between these variables?
b. Compute column percents for the table to determine the pattern of the relationship. Which region is most likely to favor the legalization of marijuana?

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Problem 14

A researcher is concerned with the relationship between attitudes toward violence and violent behavior. If attitudes "cause" behavior (a very debatable proposition), then people who have positive attitudes toward violence should have high rates of violent behavior. A pretest was conducted on 70 respondents; among other things, the respondents were asked, "Have you been involved in a violent incident of any kind over the past six months?" The researcher established the following relationship: However, the chi square obtained is a very healthy $23.4$ (confirm with your own calculations). Why is the full-sample chi square significant when the pretest was not? What happened? Do you think that the second result is important?

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Problem 15

Some results from a survey administered to a nationally representative sample are presented here. For each table, conduct the chi square test of significance and compute column percents. Write a sentence or two of interpretation for each test.
a. Support for the legal right to an abortion by age: b. Support for the death penalty by age:
c. Fear of walking alone at night by age:

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Problem 16

(This problem is optional.) The director of athletics at the local high school wonders if the sports program is getting a proportional amount of support from each of the four classes. If there are roughly equal numbers of students in each of the classes, what does the following breakdown of attendance figures from a random sample of students in attendance at a recent basketball game suggest?

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Problem 17

$($ This problem is optional. $)$ A small western town has roughly equal numbers of Hispanic, Asian, and Anglo-American residents. Are the three groups equally represented at town meetings? The attendance figures for a random sample drawn from those attending a meeting wereIs there a statistically significant pattern here?

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