Clarifying the Concepts
Calculating the Statistics
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.
| Participant | Variable X | Variable Y |
|---|---|---|
| 1 | 134.5 | 64.00 |
| 2 | 186 | 60.00 |
| 3 | 157 | 61.50 |
| 4 | 129 | 66.25 |
| 5 | 147 | 65.50 |
| 6 | 133 | 62.00 |
| 7 | 141 | 62.50 |
| 8 | 147 | 62.00 |
| 9 | 136 | 63.00 |
| 10 | 147 | 65.50 |
Consider the following scale data.
| Participant | Variable X | Variable Y |
|---|---|---|
| 1 | $1250 | 25 |
| 2 | $1400 | 21 |
| 3 | $1100 | 32 |
| 4 | $1450 | 54 |
| 5 | $1600 | 38 |
| 6 | $2100 | 62 |
| 7 | $3750 | 43 |
| 8 | $1300 | 32 |
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The following fictional data represent the finishing place for runners of a 5-
| Race Rank | Hours Trained | Race Rank | Hours Trained |
|---|---|---|---|
| 1 | 25 | 6 | 18 |
| 2 | 25 | 7 | 12 |
| 3 | 22 | 8 | 17 |
| 4 | 18 | 9 | 15 |
| 5 | 19 | 10 | 16 |
Imagine that a researcher measured a group of participants at two time points. Fictional scores for these two time points appear below. Are the scores different at time 1 and time 2?
| Person | Time 1 | Time 2 |
|---|---|---|
| 1 | 56 | 83 |
| 2 | 74 | 116 |
| 3 | 81 | 96 |
| 4 | 47 | 56 |
| 5 | 78 | 120 |
| 6 | 96 | 100 |
| 7 | 72 | 71 |
Assume a group of students provides happiness ratings for how happy they feel during the school year and how happy they feel during the summer. Do happiness levels differ depending on the time of year? Fictional data appear below:
| Student | School Year | Summer |
|---|---|---|
| 1 | 7 | 4 |
| 2 | 4 | 6 |
| 3 | 5 | 5 |
| 4 | 3 | 4 |
| 5 | 4 | 8 |
| 6 | 5 | 7 |
| 7 | 3 | 2 |
| Person | Score 1 | Score 2 |
|---|---|---|
| 1 | 6 | 6 |
| 2 | 5 | 3 |
| 3 | 4 | 2 |
| 4 | 3 | 5 |
| 5 | 2 | 1 |
| 6 | 1 | 4 |
| Group 1 | Ordinal Dependent Variable | Group 2 | Ordinal Dependent Variable |
|---|---|---|---|
| 1 | 1 | 1 | 11 |
| 2 | 2.5 | 2 | 9 |
| 3 | 8 | 3 | 2.5 |
| 4 | 4 | 4 | 5 |
| 5 | 6 | 5 | 7 |
| 6 | 10 | 6 | 12 |
| Group 1 | Scale Dependent Variable | Group 2 | Scale Dependent Variable |
|---|---|---|---|
| 1 | 8 | 9 | 3 |
| 2 | 5 | 10 | 4 |
| 3 | 5 | 11 | 2 |
| 4 | 7 | 12 | 1 |
| 5 | 10 | 13 | 1 |
| 6 | 14 | 14 | 5 |
| 7 | 9 | 15 | 6 |
| 8 | 11 |
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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:
| Student | Gender | Class Standing | Student | Gender | Class Standing |
|---|---|---|---|---|---|
| 1 | Male | 98 | 7 | Male | 43 |
| 2 | Female | 72 | 8 | Male | 33 |
| 3 | Male | 15 | 9 | Female | 17 |
| 4 | Female | 3 | 10 | Female | 82 |
| 5 | Female | 102 | 11 | Male | 63 |
| 6 | Female | 8 | 12 | Male | 25 |
Assume a researcher compared the performance of two independent groups of participants on an ordinal variable using the Mann–
| Group 1 | Group 2 | Group 3 |
|---|---|---|
| 0.22 | 1.03 | 0.52 |
| 0.55 | 0.89 | 0.67 |
| 1.20 | 0.74 | 2.83 |
| 0.83 | 1.86 | 3.20 |
| 1.01 | 2.21 | 2.75 |
| 0.86 | 0.94 | 1.74 |
| Group 1 | Group 2 | Group 3 |
|---|---|---|
| 1 | 1 | 5 |
| 5 | 3 | 4 |
| 3 | 3 | 1 |
| 2 | 4 | 1 |
| 2 | 2 | 3 |
The following data set represents the scores of three independent groups of participants on a single scale dependent variable:
| Group 1 | Group 2 | Group 3 |
|---|---|---|
| 15 | 38 | 12 |
| 27 | 22 | 72 |
| 16 | 56 | 84 |
| 41 | 33 |
Applying the Concepts
University students, cell phone bills, and ordinal data: Here are some monthly cell phone bills, in dollars, for university students:
| 100 | 60 | 35 | 50 | 50 | 50 | 60 | 65 |
| 0 | 75 | 100 | 55 | 50 | 40 | 80 | |
| 200 | 30 | 50 | 108 | 500 | 100 | 45 | |
| 40 | 45 | 50 | 40 | 40 | 100 | 80 |
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 (https:/
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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-
| Team | Fantasy League Points | Actual American League Finish |
|---|---|---|
| Boston | 117.5 | 2 |
| New York | 109.5 | 1 |
| Anaheim | 108 | 3.5 |
| Minnesota | 97 | 3.5 |
| Texas | 85 | 6 |
| Chicago | 80 | 7 |
| Cleveland | 79 | 8 |
| Oakland | 77 | 5 |
| Baltimore | 74.5 | 9 |
| Detroit | 68.5 | 10 |
| Seattle | 51 | 13 |
| Tampa Bay | 47.5 | 11 |
| Toronto | 35.5 | 12 |
| Kansas City | 20 | 14 |
Test-
98 74 87 92 88 93 62 67
Test-
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Hockey wins and the Wilcoxon signed-
| Team | 1995– |
2005– |
|---|---|---|
| Calgary Flames | 34 | 46 |
| Edmonton Oilers | 30 | 41 |
| Montréal Canadiens | 40 | 42 |
| Ottawa Senators | 18 | 52 |
| Toronto Maple Leafs | 34 | 41 |
| Vancouver Canucks | 32 | 42 |
Online plane tickets and the Wilcoxon signed-
| Itinerary | CheapTickets | Expedia |
|---|---|---|
| Athens, GA, to Johannesburg, South Africa | $2403 | $2580 |
| Chicago to Chennai, India | 1884 | 2044 |
| Columbus, OH, to Belgrade, Serbia | 1259 | 1436 |
| Denver to Geneva, Switzerland | 1392 | 1412 |
| Montréal to Dublin, Ireland | 1097 | 1152 |
| New York City to Reykjavik, Iceland | 935 | 931 |
| San Antonio to Hong Kong | 1407 | 1400 |
| Toronto to Istanbul, Turkey | 1261 | 1429 |
| Tulsa to Guadalajara, Mexico | 565 | 507 |
| Vancouver to Melbourne, Australia | 1621 | 1613 |
Public versus private universities and the Mann–
| University | Rank | Type of School | Public Rank | Private Rank |
|---|---|---|---|---|
| Princeton University | 2 | Private | 2 | |
| University of California, Berkeley | 2 | Public | 2 | |
| University of Wisconsin, Madison | 2 | Public | 2 | |
| Stanford University | 4.5 | Private | 4.5 | |
| University of Michigan, Ann Arbor | 4.5 | Public | 4.5 | |
| Harvard University | 7 | Private | 7 | |
| University of Chicago | 7 | Private | 7 | |
| University of North | 7 | Public | 7 | |
| Carolina, Chapel Hill | ||||
| University of California, Los Angeles | 9 | Public | 9 | |
| Northwestern University | 10.5 | Private | 10.5 | |
| University of Pennsylvania | 10.5 | Private | 10.5 | |
| Columbia University | 12.5 | Private | 12.5 | |
| Indiana University, Bloomington | 12.5 | Public | 12.5 | |
| Duke University | 14.5 | Private | 14.5 | |
| University of Texas, Austin | 14.5 | Public | 14.5 | |
| New York University | 16 | Private | 16 | |
| Cornell University | 18 | Private | 18 | |
| Ohio State University | 18 | Public | 18 | |
| Pennsylvania State University, University Park | 18 | Public | 18 |
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Gender, aggression, and the interpretation of a Mann–
Cell phone bill, hours studied, 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.
Angelman syndrome and bootstrapping: Angelman syndrome is a rare genetic disease in which children are delayed developmentally and exhibit unusual symptoms such as inappropriate and prolonged laughter, difficulty in speaking or inability to speak, and seizures. Imagine that a researcher obtained vocabulary data for six children with Angelman syndrome and wants to develop an estimate of the mean vocabulary score of the population of children with Angelman syndrome. (Although those with Angelman syndrome often cannot speak, they are usually able to understand at least some simple language and they may learn to communicate with sign language.) The General Social Survey (GSS), using a multiple-
Putting It All Together
“Smart” states and the Kruskal–
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We marked states in the Northeast with an NE, in the Midwest with a MW, and in the South with an S. Do these regions tend to have different rankings from one another?
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-