Distinguish among nominal, ordinal, and scale data.

What are the three main situations in which we use a nonparametric test?

What is the difference between the chi-square test for goodness of fit and the chi-square test for independence?

What are the four assumptions for the chi-square tests?

List two ways in which statisticians use the word independence or independent with respect to concepts introduced earlier in this book. Then describe how independence is used by statisticians with respect to chi square.

What are the hypotheses when conducting the chi-square test for goodness of fit?

How are the degrees of freedom for the chi-square hypothesis tests different from those of most other hypothesis tests?

Why is there just one critical value for a chi-square test, even when the hypothesis is a two-tailed test?

What information is presented in a contingency table in the chi-square test for independence?

What measure of effect size is used with chi square?

Define the symbols in the following formula:

What is the formula used for?

What information does the measure of relative likelihood provide?

In order to calculate relative likelihood, what must we first calculate?

What is the difference between relative likelihood and relative risk?

Why might relative likelihood be easier to understand as an effect size than Cramér’s V ?

Which graph is most useful for displaying the results of a chi-square test for independence?

Why can relative likelihood and relative risk sometimes exaggerate risks? (Hint: Think about the base rates—how common a disorder or other occurrence is to start with.)

When do we convert scale data to ordinal data?

When the data on at least one variable are ordinal, the data on any scale variable must be converted from scale to ordinal. How do we convert a scale variable into an ordinal one?

How does the transformation of scale data to ordinal data solve the problem of outliers?

What does a histogram of rank-ordered data look like and why does it look that way?

Explain how the relation between ranks is the core of the Spearman rank-order correlation.

Define the symbols in the following term:

What is the possible range of values for the Spearman rank-order correlation and how are these values interpreted?

How would you respond in a situation in which you are ranking a set of scale data and there are two numbers that are exactly the same?

When is it appropriate to use the Wilcoxon signed-rank test?

When do we use the Mann–Whitney U test?

What are the assumptions of the Mann–Whitney U test?

How are the critical values for the Mann–Whitney U test used differently than critical values for parametric tests?

If the data meet the assumptions of the parametric test, why is it preferable to use the parametric test rather than the nonparametric alternative?

For each of the following, (i) identify the incorrect symbol, (ii) state what the correct symbol should be, and (iii) explain why the initial symbol was incorrect.

For each of the following, identify the independent variable(s), the dependent variable(s), and the level of measurement (nominal, ordinal, scale).

Use this calculation table for the chi-square test for goodness of fit to complete this exercise.

Use this calculation table for the chi-square test for goodness of fit to complete this exercise.

Below are some data to use in a chi-square test for independence.

The data below are from a study of lung cancer patients in Turkey (Yilmaz et al., 2000). Use these data to calculate the relative likelihood of the patient being a smoker, given that a person is female rather than male.

In order to compute statistics, we need to have working formulas. For the following, (i) identify the incorrect symbol, (ii) state what the correct symbol should be, and (iii) explain why the initial symbol was incorrect.

Consider the following scale data.

Consider the following scale data.

The following fictional data represent the finishing place for runners of a 5-kilometer race and the number of hours they trained per week.

Compute the Mann–Whitney U statistic for the following data. Each participant has been assigned a group and a participant number; these are shown in the “Group 1” and “Group 2” columns.

Compute the Mann–Whitney U statistic for the following data. Each participant has been assigned a group and a participant number; these are shown in the “Group 1” and “Group 2” columns.

Assume a researcher compared the performance of two independent groups of participants on an ordinal variable using the Mann–Whitney U test. The first group had 8 participants and the second group had 11 participants.

Are men or women more likely to be at the top of their class? The following table depicts fictional class standings for a group of men and women:

A nonparametric test and gender differences in obeying bicycling laws: Students at Hunter College studied bicycle safety in New York City (Tuckel & Milczarski, 2014). They reported data on cyclists who were riding their own bikes and were not cycling as part of their job (they were not, for example, riding as delivery workers). They reported that 28.4% of male cyclists stopped completely at a red light, whereas 38.3% of female cyclists did so. 30.9% of male cyclists and 35.4% of female cyclists paused but then rolled through the red light. And 40.7% of male cyclists and 26.3% of female cyclists just ran the light. What hypothesis test would the researchers conduct? Explain your answer.

Gender, salary negotiation, and chi square: Researchers investigated whether or not language in job postings affected the likelihood that women and men would negotiate regarding salary (Leibbrandt & List, 2012). Some job postings clearly indicated that the salary was negotiable, and others contained no such statement. The postings were otherwise identical. The researchers examined the behavior of almost 2500 applicants for one of the jobs in these advertisements. The graph shows the proportions of women and men who negotiated in response to either type of listing.

Gender, the Oscars, and nonparametric tests: In 2010, Sandra Bullock won an Academy Award for best actress. Shortly thereafter, she discovered that her husband was cheating on her. Headlines erupted about a supposed Oscar curse that befalls women, and many in the media wondered whether ambitious women—whether actors or corporate leaders—are more likely than ambitious men to run the risk of ruining their family lives. Reporters breathlessly listed female actors who were divorced within a couple of years of winning an Oscar—Julia Roberts, Helen Hunt, Kate Winslet, Halle Berry, and Reese Witherspoon among them.

Parametric or nonparametric test? For each of the following research questions, state whether a parametric or nonparametric hypothesis test is more appropriate. Explain your answers.

Types of variables and student evaluations of professors: Weinberg, Fleisher, and Hashimoto (2007) studied almost 50,000 students’ evaluations of their professors in nearly 400 economics courses at the Ohio State University over a 10-year period. For each of their findings, outlined below, state (i) the independent variable or variables, and, where appropriate, their levels; (ii) the dependent variable(s); and (iii) which category of research design is being used:

I—Scale independent variable(s) and scale dependent variable

II—Nominal independent variable(s) and scale dependent variable

III—Only nominal variables

Explain your answer to part (iii).

Grade inflation and types of variables: A New York Times article on grade inflation reported several findings related to a tendency for average grades to rise over the years and a tendency for the top-ranked institutions to give the highest average grades (Archibold, 1998). For each of the findings outlined below, state (i) the independent variable or variables, and, where appropriate, their levels; (ii) the dependent variable(s); and (iii) which category of research design is being used:

I—Scale independent variable(s) and scale dependent variable

II—Nominal independent variable(s) and scale dependent variable

III—Only nominal variables

Explain your answer to part (iii).

High school academic performance and types of variables: Here are three ways to assess one’s performance in high school: (1) GPA at graduation, (2) whether one graduated with honors (as indicated by graduating with a GPA of at least 3.5), and (3) class rank at graduation. For example, Abdul had a 3.98 GPA, graduated with honors, and was ranked 10^th in his class.

Immigration, crime, and research design: “Do Immigrants Make Us Safer?” asked the title of a New York Times Magazine article (Press, 2006). The article reported findings from several U.S.-based studies, including several conducted by Harvard sociologist Robert Sampson in Chicago. For each of the following findings, draw the table of cells that would comprise the research design. Include the labels for each row and column.

Sex selection and hypothesis testing: Across all of India, there are only 933 girls for every 1000 boys (Lloyd, 2006), evidence of a bias that leads many parents to illegally select for boys or to kill their infant girls. (Note that this translates into a proportion of girls of 0.483.) In Punjab, a region of India in which residents tend to be more educated than in other regions, there are only 798 girls for every 1000 boys. Assume that you are a researcher interested in whether sex selection is more or less prevalent in educated regions of India, and that 1798 children from Punjab constitute the entire sample. (Hint: You will use the proportions from the national database for comparison.)

Gender, op-ed writers, and hypothesis testing: Richards (2006) reported data from a study by the American Prospect on the genders of op-ed writers who addressed the topic of abortion in the New York Times. Over a 2-year period, the American Prospect counted 124 articles that discussed abortion (from a wide range of political and ideological perspectives). Of these, just 21 were written by women.

Romantic music, behavior, chi square, and effect size: Guéguen, Jacob, and Lamy (2010) investigated whether exposure to romantic music affects dating behavior. The participants, young, single French women, waited for the experiment to start in a room in which songs with either romantic lyrics or neutral lyrics were playing. After a few minutes, each woman who participated completed a marketing survey administered by a young male confederate. During a break, the confederate asked the participant for her phone number. Of the women who listened to romantic music, 52.2% (23 out of 44) gave him her phone number, whereas 27.9% (12 out of 43) of the women who listened to neutral music did so. The researchers conducted a chi-square test for independence, and found the following results: (χ² (1, N = 83) = 5.37, p < .02).

The General Social Survey, an exciting life, and relative risk: In How It Works 15.2, we walked through a chi-square test for independence using two items from the General Social Survey (GSS)—LIFE and MOBILE16. Use these data to answer the following questions.

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Gender, ESPN, and chi square: Many of the numbers we see in the news could be analyzed with chi square. The feminist blog Culturally Disoriented examined the photos in the 2012 “body issue” of ESPN The Magazine—the publication’s annual spread of photographs of nude athletes. The blogger reported: “Female athlete after female athlete was photographed not as a talented, powerful sportswoman, but as. . .eye candy” (http://culturallydisoriented.wordpress.com/2012/07/12/the-bodies-we-want-female-athletes-in-espn-magazines-body-issue/). The blogger reported that there were 19 photos of male athletes and 17 of female athletes. Of these, 15 of the men were in active poses and 9 of the women were in active poses. Active poses were typically those in which they were engaged in their sport, whereas the passive photos looked more like a modeling shoot—“where they’re just looking hot for the camera,” in the words of Culturally Disoriented.

Premarital doubts, divorce, and chi square: In an article titled “Do Cold Feet Warn of Trouble Ahead?”, researchers studied 464 married heterosexual spouses to determine whether or not doubts before marriage were predictive of marital troubles, and divorce, later on (Lavner, Karney, & Bradbury, 2012). The following is an excerpt from the results section of their paper: “For husbands, 9% of those who reported not having premarital doubts divorced by four years (n = 10 of 117) compared with 14% of those who did report premarital doubts (n = 15 of 106); these groups did not differ significantly, χ² (1, n = 223) = 1.76, p > .10. Among wives, 8% of those who reported not having premarital doubts divorced by four years (n = 11 of 141) compared with 19% of those who did report premarital doubts (n = 16 of 84). Chi-square analyses indicated that these rates differed significantly, χ² (1, n = 225) = 6.31, p < .05.”

University students, cell phone bills, and ordinal data: Here are some monthly cell phone bills, in dollars, for university students:

World cities, livability, and nonparametric hypothesis tests: CNN.com reported on a 2012 study that ranked the world’s cities in terms of how livable they are (http://travel.cnn.com/explorations/life/worlds-most-livable-city-525619), using a range of criteria related to stability, health care, culture and environment, education, and infrastructure. The top 10, in order, were: Melbourne, Australia; Vienna, Austria; Vancouver, Toronto, and Calgary, all in Canada; Adelaide and Sydney, both in Australia; Helsinki, Finland; Perth, Australia; and Auckland, New Zealand. For each of the following research questions, state which nonparametric hypothesis test is appropriate: the Spearman rank-order correlation coefficient, the Wilcoxon signed-rank test, the Mann–Whitney U test, or the Kruskal–Wallis H test. Explain your answers and indicate the equivalent parametric test.

Fantasy baseball and the Spearman correlation coefficient: In fantasy baseball, groups of 12 league participants conduct a draft in which they can “buy” any baseball players from any teams across one of the two Major League Baseball (MLB) leagues (the American League and the National League). These makeshift teams are compared on the basis of the combined statistics of the individual baseball players. For example, statistics about home runs are transformed into points, and each fantasy team receives a total score of all combined points based on its baseball players, regardless of their real-life team. Many in the fantasy and real-life baseball worlds have wondered how success in fantasy leagues maps onto real MLB teams’ success in terms of winning baseball games. Walker (2006) compared the fantasy league performances of the players for each American League team with their actual American League finishes for the 2004 season, the year the Boston Red Sox broke the legendary “curse” against them and won the World Series. The data, sorted from highest to lowest fantasy league score, are shown in the accompanying table.

Test-taking speed, grade, and the Spearman correlation coefficient: Does speed in completing a test correlate with one’s grade? Here are test scores for eight students in one of our statistics classes. They are arranged in order from the student who turned in the test first to the student who turned in the test last.

1.00, −0.001, 0.52, −0.27, −0.98, 0.09

Specify which of these coefficients suggests the strongest relation between the two variables as well as which coefficient suggests the weakest relation between the two variables.

Nonparametric tests and a longitudinal study of mathematically gifted teenagers: Researchers studied male and female teenagers who had scored in the top 1% on mathematical reasoning measures (Lubinski, Benbow, & Kell, 2014). The researchers were able to follow the participants’ progression over the next 40 years. They tracked two groups, or cohorts, of participants on a number of variables, including career-related variables. They reported “pronounced and significant sex differences in the percentage of participants who were working full time (Cohort 1: 89% of males and 69% of females; Cohort 2: 90% of males and 59% of females), χ²(1, N = 1,131) = 67.56, p < .001, and χ²(1, N = 481) = 68.55, p < .001.” So, the researchers looked at income only among those who were employed full time. For example, for one cohort they reported median incomes of $150,000 for men and $101,000 for women. This gender difference was statistically significant for both groups: “Mann–Whitney U test, zs ≥ 5.09, ps < .001.”

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Public versus private universities and the Mann–Whitney U test: Do public or private universities tend to have better sociology graduate programs? U.S. News & World Report publishes online rankings of graduate schools across a range of disciplines. The table below includes the Report’s 2013 list of the top 19 doctoral programs in sociology and notes whether the schools are public or private. Schools listed at the same rank are tied.

Gender, aggression, and the interpretation of a Mann–Whitney U test: Spanish researchers examining aggression in children’s dreams reported the following: “Using the Mann–Whitney nonparametrical statistical test on the gender differences, we found a significant difference between boys and girls in Group 1 for overall [aggression] (U = 44.00, p = 0.004) and received aggression (U = 48.00, p = 0.005). So, in their dreams, younger boys not only had a higher level of general aggression but also received more severe aggressive acts than girls of the same age” (emphasis in original) (Oberst, Charles, & Chamarro, 2005, p. 175).

Cell phone bill, hours spent studying, and the shapes of distributions: The following figures display data that depict the relation between students’ monthly cell phone bills and the number of hours they report that they study per week.

Stroke patients, treatment, and type of nonparametric test: A common situation faced by researchers working with special populations, such as neurologically impaired people or people with less common psychiatric conditions, is that the studies often have small sample sizes due to the relatively few number of patients. As a result, these researchers often turn to nonparametric statistical tests. For each of the following research descriptions, state which nonparametric hypothesis test is most appropriate: the Spearman rank-order correlation coefficient, the Wilcoxon signed-rank test, the Mann–Whitney U test, or the Kruskal–Wallis H test. Explain your answers.

Gender bias, poor growth, and hypothesis testing: Grimberg, Kutikov, and Cucchiara (2005) wondered whether gender biases were evident in referrals of children for poor growth. They believed that boys were more likely to be referred even when there was no problem—which is bad for boys because families of short boys might falsely view their height as a medical problem. They also believed that girls were less likely to be referred even when there was a problem—which is bad for girls because real problems might not be diagnosed and treated. They studied all new patients at the Children’s Hospital of Philadelphia Diagnostic and Research Growth Center who were referred for potential problems related to short stature. Of the 182 boys who were referred, 27 had an underlying medical problem, 86 did not but were below norms for their age, and 69 were of normal height according to growth charts. Of the 96 girls who were referred, 39 had an underlying medical problem, 38 did not but were below norms for their age, and 19 were of normal height according to growth charts.

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The prisoner’s dilemma, cross-cultural research, and hypothesis testing: In a classic prisoner’s dilemma game with money for prizes, players who cooperate with each other both earn good prizes. If, however, your opposing player cooperates but you do not (the term used is defect), you receive an even bigger payout and your opponent receives nothing. If you cooperate but your opposing player defects, he or she receives that bigger payout and you receive nothing. If you both defect, you each get a small prize. Because of this, most players of such games choose to defect, knowing that if they cooperate but their partners don’t, they won’t win anything. The strategies of U.S. and Chinese students were compared. The researchers hypothesized that those from the market economy (United States) would cooperate less (i.e., would defect more often) than would those from the nonmarket economy (China).