Skip to content
Textbook Guides

Statistics for the Behavioral Sciences Third Edition Solution Manual

Statistics for the Behavioral Sciences Third Edition Solution Manual is your guide to textbook mastery, offering detailed solutions to every chapter's exercises.

Preview 16 of 140 pages

See the first 16 pages exactly as they appear in the document.

  1. Page 1
    Read page 1 as text

    C-1 C H A P T E R 1 1.1 Descriptive statistics organize, summarize, and communicate a group of numerical observations. Inferential statistics use sample data to make general estimates about the larger population. 1.2 A sample is a set of observations drawn from the population of interest that, it is hoped, shares the same characteristics as the population of interest. A population includes all possible observations about which we’d like to know something. 1.3 The four types of variables are nominal, ordinal, interval, and ratio. A nominal variable is used for observations that have categories, or names, as their values. An ordinal variable is used for observations that have rankings (i.e., 1st, 2nd, 3rd) as their values. An interval variable has numbers as its values; the distance (or interval) between pairs of consecutive numbers is assumed to be equal. Finally, a ratio variable meets the criteria for interval variables but also has a meaningful zero point. Interval and ratio variables are both often referred to as scale variables. 1.4 Statisticians use scale as another term for an interval or ratio measure. They also use scale as a word for many measurement tools, particularly those that involve a series of items that test- takers must complete. 1.5 Discrete variables can only be represented by specific numbers, usually whole numbers; continuous variables can take on any values, including those with great decimal precision (e.g., 1.597). 1.6 An independent variable is a variable that we either manipulate or observe to determine its effects on the dependent variable; a dependent variable is the outcome variable that we hypothesize to be related to, or caused by, changes in the independent variable. 1.7 A confounding variable (also called a confound) is any variable that systematically varies with the independent variable so that we cannot logically determine which variable affects the dependent variable. Researchers attempt to control confounding variables in experiments by randomly assigning participants to conditions. The hope with random assignment is that the confounding variable will be spread equally across the different conditions of the study, thus neutralizing its effects. 1.8 Reliability refers to the consistency of a measure. Validity refers to the extent to which a test actually measures what it was intended to measure. A measure that is valid absolutely must be reliable, but a reliable measure is not necessarily a valid one. 1.9 An operational definition specifies the operations or procedures used to measure or manipulate an independent or dependent variable. 1.10 In everyday language, people often use the word experiment to refer to something they are trying out to see what will happen. Researchers use the term to refer to a type of study in which participants are randomly assigned to levels of the independent variable. 1.11 When conducting experiments, the researcher randomly assigns participants to conditions or levels of the independent variable. When random assignment is not possible, such as when studying something like gender or marital status, correlational research is used. Correlational research allows us to examine how variables are related to each other; experimental research allows us to make assertions about how an independent variable causes an effect in a dependent variable. 1.12 In a between-groups research design, participants experience one, and only one, level of the independent variable. In a within-groups research design, all levels of the independent variable are experienced by all participants in the study. 1.13 a. “This was an experiment.” (not “This was a correlational study.”) b. “. . . the independent variable of caffeine . . .” (not “ . . . the dependent variable of caffeine . . . ”) c. “A university assessed the validity . . .” (not “A university assessed the reliability . . . ”) d. “In a between-groups experiment . . .” (not “In a within- groups experiment . . . ”) 1.14 a. “. . . the nominal variable ‘gender’ . . .” (not “. . . the ordinal variable ‘gender’ . . .”) b. “A psychologist used a within-groups design . . .” (not “A psychologist used a between-groups design . . .”) c. “. . . the effects of the independent variable . . .” (not “. . . the effects of the confounding variable . . .”) d. “A researcher studied a sample of 20 rats . . .” (not “A researcher studied a population of 20 rats . . .”) 1.15 a. An outlier is a participant or observation that is very different from other observations in the study. b. When identifying why a particular observation is so different from the other observations in the study (i.e., outlier analysis), the researcher may gain insight into other factors that influence the dependent variable. Solutions to End-of-Chapter Problems a p p e n d i x c Nolan3e_interior_Appendix_C_Ch01.indd 1 11/04/14 6:53 PM

  2. Page 2
  3. Page 3
    Read page 3 as text

    C-2 APPENDIX C 1.16 The sample is the 225 students who completed the survey. The population is all of the student customers at the bookstore. 1.17 The sample is the 100 customers who completed the survey. The population is all of the customers at the grocery store. 1.18 a. 130 people b. All people living in urban areas in the United States c. Descriptive statistic d. Answers may vary, but one way is to sort people into groups such as “long distance walked,” “medium distance walked,” and “short distance walked.” e. Answers may vary, but pedometers could be used to measure steps taken or miles walked, both of which are scale measures. 1.19 a. 73 people b. All people who shop in grocery stores similar to the one where data were collected c. Inferential statistic d. Answers may vary, but people could be labeled as having a “healthy diet” or an “unhealthy diet.” e. Answers may vary, but there could be groupings such as “no items,” “a minimal number of items,” “some items,” and “many items.” f. Answers may vary, but the number of items could be counted or weighed. 1.20 Answers may vary, but on a national level, one could look at the rate of houses in foreclosure or the amount of government debt. 1.21 a. The independent variables are physical distance and emotional distance. The dependent variable is accuracy of memory. b. There are two levels of physical distance (within 100 miles and 100 miles or farther) and three levels of emotional distance (knowing no one who was affected, knowing people who were affected but lived, and knowing someone who died). c. Answers may vary, but accuracy of memory could be operationalized as the number of facts correctly recalled. 1.22 a. Skin tone b. Severity of facial wrinkles c. Three levels (light, medium, and dark) 1.23 Both Miguel Induráin and Lance Armstrong could be considered outliers because their scores (number of wins) are extreme compared to the typical number of wins by Tour de France winners. 1.24 An outlier analysis of Miguel Induráin and Lance Armstrong might lead a researcher to identify factors that are critical to superior cycling performance. 1.25 a. The average weight for a 10-year-old girl was 77.4 pounds in 1963 and nearly 88 pounds in 2002. b. No; the CDC would not be able to weigh every single girl in the United States because it would be too expensive and time consuming. c. It is a descriptive statistic because it is a numerical summary of a sample. It is an inferential statistic because the researchers drew conclusions about the population’s average weight based on this information from a sample. 1.26 a. The sample is the 60,000 people they studied. b. The researchers would like to generalize their findings to the population of all Norwegians, or perhaps even more broadly. 1.27 a. Ordinal b. Scale c. Nominal 1.28 a. Ordinal b. Scale c. Scale d. Scale e. Nominal f. Nominal 1.29 a. Discrete b. Continuous c. Discrete d. Discrete e. Continuous 1.30 a. A reliable test is one that provides consistent results. If you take the test twice, you should get the same results, an indication of reliability. b. A valid test is one that measures what it intends to measure. This test has the stated intention of measuring personality. If in fact it is measuring personality accurately, then it is a valid test. c. There are several possible answers to this question. The developers of this Web site might, for example, hypothesize that the region of the world in which one grew up predicts different personality profiles that are based on region. d. The independent variable would be region and the dependent variable would be personality profile. 1.31 a. The independent variables are temperature and rainfall. Both are continuous scale variables. b. The dependent variable is experts’ ratings. This is a discrete scale variable. c. The researchers wanted to know if the wine experts are consistent in their ratings—that is, if they’re reliable. d. This observation would suggest that Robert Parker’s judgments are valid. His ratings seem to be measuring what they intend to measure—wine quality. 1.32 a. “Best rapper” is operationalized as those who have the highest rhyme density. b. Other variables could include ratings of flow and delivery when performing, measures of influence or impact, ratings of storytelling and lyricism, record or song sales, awards won, number of downloads, critic reviews and ratings, listener ratings, and so on. c. Ranking is an ordinal variable; rhyme density is a scale variable. d. They added cultural relevance and how interesting the rapper is to their operational definition. 1.33 a. Age: teenagers and adults in their 30s; video game performance: final score on a video game or average reaction time on a video game task b. Spanking: spanking and not spanking; violent behavior: parental measure of child aggression or number of aggressive acts observed in an hour of play. c. Meetings: go to meetings and participate online; weight loss: measured in pounds or kilograms, or by change in waist size Nolan3e_interior_Appendix_C_Ch01.indd 2 11/04/14 6:53 PM

  4. Page 4
    Read page 4 as text

    APPENDIX C C-3 d. Studying: with others and alone; statistics performance: average test score for the semester or overall grade for the semester. e. Beverage: caffeinated and decaffeinated; time to fall asleep: minutes to fall asleep from when the participant goes to bed, or the actual time at which the participant falls asleep. 1.34 a. The study could use a between-groups research design by assigning half the participants to exercise and half not to exercise. b. Participants could be followed for several months to determine weight loss before the exercise program, then start the exercise program and be followed for several months to determine weight loss after the program. c. There are several possible confounds. In the within-groups design, the participants are having their weight loss tracked, then starting an exercise program, then having their weight loss tracked some more. It is possible that the mere act of tracking weight loss leads participants to implement weight-loss tactics other than exercise and that they start reaping the benefits of these tactics around the time the exercise program begins. Alternatively, it is possible that the no exercise segment occurs in the winter and the exercise segment occurs in the spring. Many people gain a bit of weight during the winter and lose weight as summer—and bathing-suit season—approaches. It might be the weather, not the exercise program, that leads to weight loss. 1.35 a. An experiment requires random assignment to conditions. It would not be ethical to randomly assign some people to smoke and some people not to smoke, so this research had to be correlational. b. Other unhealthy behaviors have been associated with smoking, such as poor diet and infrequent exercise. These other unhealthy behaviors might be confounded with smoking. c. The tobacco industry could claim it was not the smoking that was harming people, but rather the other activities in which smokers tend to engage or fail to engage. d. You could randomly assign people to either a smoking group or a nonsmoking group, and assess their health over time. 1.36 a. This research is correlational because participants could not be randomly assigned to be high in individualism or collectivism. b. The sample is the 32 people who tested high for individualism and the 37 people who tested high for collectivism. c. Answers may vary, but one hypothesis could be “On average, people high in individualism will have more relationship conflict than those high in collectivism.” d. Answers may vary, but one way to measure relationship conflict could be counting the number of disagreements or fights per month. 1.37 a. This is experimental because students are randomly assigned to one of the incentive conditions for recycling. b. Answers may vary, but one hypothesis could be “Students fined for not recycling will report lower concerns for the environment, on average, than those rewarded for recycling.” 1.38 a. The person would be considered an outlier because his or her score was far from the scores of all the others in the study. b. Outlier analysis might be useful to find out why this person gained so much despite the exercise program. For example, was the person eating much more because he or she incorrectly assumed the exercise would burn all the extra calories? c. We are looking for any reason that might explain why this outlier exists. Is there something about this individual that provides evidence for our hypothesis, when, on the face of it, the outlier seems to discredit our hypothesis? 1.39 a. The person who took 3 minutes would be considered an outlier because the person’s response time was much more extreme than any of the response times exhibited by the other participants. b. In this case, the researcher might look to see if the participant was slow on other experimental tasks as well or if there was some other independent evidence that the participant did not take the experimental task seriously. 1.40 a. Participants in the Millennium Cohort Study b. Parents in the United Kingdom, or possibly all parents globally c. This is a correlational study, as individuals were not randomly assigned to the condition of being a married couple or a cohabiting couple. d. Marital status—married or cohabiting e. Length of relationship f. There are several possible answers to this question. For example, economic status or financial well-being may be a confounding factor, as those who are more likely to have the money to marry and raise a family may have fewer life stressors than those who have less money, do not marry, and choose to cohabit. This variable could be operationalized and measured via household income. 1.41 a. Researchers could have randomly assigned some people who are HIV-positive to take the oral vaccine and other people who are HIV-positive not to take the oral vaccine. The second group would likely take a placebo. b. This would have been a between-groups experiment because the people who are HIV-positive would have been in only one group: either vaccine or no vaccine. c. This limits the researchers’ ability to draw causal conclusions because the participants who received the vaccine may have been different in some way from those who did not receive the vaccine. There may have been a confounding variable that led to these findings. For example, those who received the vaccine might have had better access to health care and better sanitary conditions to begin with, making them less likely to contract cholera regardless of the vaccine’s effectiveness. d. The researchers might not have used random assignment because it would have meant recruiting participants, likely immunizing half, then following up with all of them. The researchers likely did not want to deny the vaccine to people who were HIV-positive because they might have contracted cholera and died without it. 1.42 a. Ability level, graduate level (high school versus university), race b. Wages c. 12,000 men and women in the United States who were 14–22 years old in 1979 d. High school and college graduate men and women in the United States e. Participants were studied over time to measure change during that period. f. Age could be a confounding variable, as those who are older will have greater exposure to the various areas measured via the AFQT, in addition to the education they received at the college level. Nolan3e_interior_Appendix_C_Ch01.indd 3 11/04/14 6:53 PM

  5. Page 5
    Read page 5 as text

    C-4 APPENDIX C g. Ability could be operationalized by having managers rate each participant’s ability to perform his or her job. Another way ability could be operationalized is via high school and college GPA or a standardized ability test. 1.43 a. A “good charity” is operationally defined as one that spends more of its money for the cause it is supporting and less for fundraising or administration. b. The rating is a scale variable, as it has a meaningful zero point, has equal distance between intervals, and is continuous. c. The tier is an ordinal variable, as it involves ranking the organizations into categories (1st, 2nd, 3rd, 4th, or 5th tier) and it is discrete. d. The type of charity is a nominal variable, as it uses names or categories to classify the values (e.g., health and medical needs) and it is discrete. e. Measuring finances is more objective and easier to measure than some of the criteria mentioned by Ord, such as importance of the problem and competency and honesty. f. Charity Navigator’s ratings are more likely to be reliable than GiveWell’s ratings because they are based on an objective measure. It is more likely that different assessors would come up with the same rating for Charity Navigator than for GiveWell. g. GiveWell’s ratings are likely to be more valid than Charity Navigator’s, provided that they can attain some level of reliability. GiveWell’s more comprehensive rating system incorporates a better-rounded assessment of a charity. h. This would be a correlational study because donation funds, the independent variable, would not be randomly assigned based on country but measured as they naturally occur. i. This would be an experiment because the levels of donation funds, the independent variable, are randomly assigned to different regions to determine the effect on death rate. C H A P T E R 2 2.1 Raw scores are the original data, to which nothing has been done. 2.2 To create a frequency table: (1) Determine the highest and lowest scores. (2) Create two columns; label the first with the variable name and label the second “Frequency.” (3) List the full range of values that encompasses all the scores in the data set, from lowest to highest, even those for which the frequency is 0. (4) Count the number of scores at each value, and write those numbers in the frequency column. 2.3 A frequency table is a visual depiction of data that shows how often each value occurred; that is, it shows how many scores are at each value. Values are listed in one column, and the numbers of individuals with scores at that value are listed in the second column. A grouped frequency table is a visual depiction of data that reports the frequency within each given interval, rather than the frequency for each specific value. 2.4 Statisticians might use interval to describe a type of variable. Interval variables have numbers as their values, and the distance (or interval) between numbers is assumed to be equal. Statisticians might also use interval to refer to the range of values to be used in a grouped frequency table, histogram, or polygon. 2.5 Bar graphs typically provide scores for nominal data, whereas histograms typically provide frequencies for scale data. Also, the categories in bar graphs do not need to be arranged in a particular order and the bars should not touch, whereas the intervals in histograms are arranged in a meaningful order (lowest to highest) and the bars should touch each other. 2.6 The x -axis is typically labeled with the name of the variable of interest. The y -axis is typically labeled “Frequency.” 2.7 A histogram looks like a bar graph but is usually used to depict scale data, with the values (or midpoints of intervals) of the variable on the x -axis and the frequencies on the y -axis. A frequency polygon is a line graph, with the x -axis representing values (or midpoints of intervals) and the y -axis representing frequencies; a dot is placed at the frequency for each value (or midpoint), and the points are connected. 2.8 Visual displays of data often help us see patterns that are not obvious when we examine a long list of numbers. They help us organize the data in meaningful ways. 2.9 In everyday conversation, you might use the word distribution in a number of different contexts, from the distribution of food to a marketing distribution.A statistician would use distribution only to describe the way that a set of scores, such as a set of grades, is distributed. A statistician is looking at the overall pattern of the data—what the shape is, where the data tend to cluster, and how they trail off. 2.10 A normal distribution is a specific frequency distribution that is a bell-shaped, symmetric, unimodal curve. 2.11 With positively skewed data, the distribution’s tail extends to the right, in a positive direction, and with negatively skewed data, the distribution’s tail extends to the left, in a negative direction. 2.12 A floor effect occurs when there are no scores below a certain value; a floor effect leads to a positively skewed distribution because the lower part of the distribution is constrained. 2.13 A ceiling effect occurs when there are no scores above a certain value; a ceiling effect leads to a negatively skewed distribution because the upper part of the distribution is constrained. 2.14 A stem-and-leaf plot retains information about every unique data point in a set, whereas a histogram does not. Additionally, it is easy to create side-by-side stem-and-leaf plots for different groups to compare their distributions. Such a side-by-side comparison of groups is not as easy to do with histograms. 2.15 A stem-and-leaf plot is much like a histogram in that it conveys how often different values in a data set occur. Also, when a stem-and-leaf plot is turned on its side, it has the same shape as a histogram of the same data set. 2.16 4.98% and 2.27% 2.17 17.95% and 40.67% 2.18 3.69% and 18.11% are scale variables, both as counts and as percentages. 2.19 0.10% and 96.77% 2.20 1,889.00, 2.65, and 0.08 2.21 0.04, 198.22, and 17.89 2.22 a. The full range is the maximum (27) minus the minimum (0), plus 1, which equals 28. b. Five c. The intervals would be 0–4, 5–9, 10–14, 15–19, 20–24, and 25–29. Nolan3e_interior_Appendix_C_Ch01.indd 4 11/04/14 6:53 PM

  6. Page 6
    Read page 6 as text

    APPENDIX C C-5 2.23 The full range of data is 68 minus 2, plus 1, or 67. The range (67) divided by the desired seven intervals gives us an interval size of 9.57, or 10 when rounded. The seven intervals are: 0–9, 10–19, 20–29, 30–39, 40–49, 50–59, and 60–69. 2.24 37.5, 52.5, and 67.5 2.25 26 shows 2.26 Twelve countries had between 2 and 10 first- or second-place World Cup finishes. 2.27 Serial killers would create positive skew, adding high numbers of murders to the data that are clustered around 1. 2.28 People convicted of murder are assumed to have killed at least one person, so observations below one are not seen, which creates a floor effect. 2.29 a. For the college population, the range of ages extends farther to the right (with a larger number of years) than to the left, creating positive skew. b. The fact that youthful prodigies have limited access to college creates a sort of floor effect that makes low scores less possible. 2.30 a. Assuming that most people go for the maximum number of friends, for the range of Facebook friends, the number of friends extends farther to the left (with fewer number of friends) than to the right, creating a negative skew. b. The fact that Facebook cuts off or limits the number of friends to 5000 means there is a ceiling effect that makes higher scores impossible. 2.31 a. The stem-and-leaf plot is depicted below: 3 55568888 3 0013334444 2 5778889 2 00344 1 688 1 03 0 5 0 1 b. This stem-and-leaf plot depicts a negatively skewed distribution. 2.32 a. The stem-and-leaf plot is depicted below: 4 00055 3 00005555555 2 00555 b. This stem-and-leaf plot depicts a symmetric distribution. 2.33 a. PERCENTAGE FREQUENCY PERCENTAGE 10 1 5.26 9 0 0.00 8 0 0.00 7 0 0.00 6 0 0.00 5 2 10.53 4 2 10.53 3 4 21.05 2 4 21.05 1 5 26.32 0 1 5.26 b. In 10.53% of these schools, exactly 4% of the students reported that they wrote between 5 and 10 twenty-page papers that year. c. This is not a random sample. It includes schools that chose to participate in this survey and opted to have their results made public. d. 1 2 3 6 8 7 9 5 4 0 0 1 3 Frequency Percent of students 5 7 9 11 e. One f. The data are clustered around 1% to 4%, with a high outlier, 10%. 2.34 a. YEARS TO COMPLETE FREQUENCY 15 2 14 1 13 1 12 1 11 1 10 2 9 4 8 9 7 11 6 10 b. 30 c. A grouped frequency table is not necessary here. These data are relatively easy to interpret in the frequency table. Grouped frequency tables are useful when the list of data is long and difficult to interpret. d. These data are clustered around 6 to 8 years, with a long tail of data out to a greater number of years to complete. These data show positive skew. Nolan3e_interior_Appendix_C_Ch02.indd 5 11/04/14 6:53 PM

  7. Page 7
    Read page 7 as text

    C-6 APPENDIX C e. 2 1 4 3 7 6 5 11 12 10 9 8 0 1 0 2 3 4 5 6 7 8 9 10 11 12 14 16 15 13 Years Frequency d. e. There are no unusual scores, as the distribution is fairly uniform with frequencies between 6 and 13. The center of the distribution seems to be in the 20–49 range. 2.36 a. INTERVAL FREQUENCY 60–69 1 50–59 5 40–49 9 30–39 5 20–29 8 10–19 2 2 1 4 3 7 6 5 11 12 14 13 10 9 8 0 0 5 15 25 35 45 55 65 75 Frequency Acceptance rate f. Eight 2.35 a. INTERVAL FREQUENCY 60–69 9 50–59 8 40–49 13 30–39 13 20–29 8 10–19 12 0–9 7 b. There are many possible answers to this question. For example, we might ask whether the prestige of the university or the region of the country is a factor in acceptance rate. c. 2 1 4 3 7 6 5 11 12 14 13 10 9 8 0 0 5 15 25 35 45 55 65 Acceptance rate Frequency Nolan3e_interior_Appendix_C_Ch02.indd 6 11/04/14 6:53 PM

  8. Page 8
    Read page 8 as text

    APPENDIX C C-7 b. 2 1 4 3 7 6 5 10 9 8 0 15 5 0 25 35 45 55 65 Number of wins Frequency c. The summary will differ for each student but should include the following information: The data appear to be roughly symmetric. d. With so few data points, it is easy to view patterns in the data without grouping the data into intervals. 2.37 a. Extroversion scores are most likely to have a normal distribution. Most people would fall toward the middle, with some people having higher levels and some having lower levels. b. The distribution of finishing times for a marathon is likely to be positively skewed. The floor is the fastest possible time, a little over 2 hours; however, some runners take as long as 6 hours or more. Unfortunately for the very, very slow but unbelievably dedicated runners, many marathons shut down the finish line 6 hours after the start of the race. c. The distribution of numbers of meals eaten in a dining hall in a semester on a three-meal-a-day plan is likely to be negatively skewed. The ceiling is three times per day, multiplied by the number of days; most people who choose to pay for the full plan would eat many of these meals. A few would hardly ever eat in the dining hall, pulling the tail in a negative direction. 2.38 a. You would present individual data values because the few categories of eye color would result in a readable list. A frequency table would be most appropriate. b. You would present grouped data because it is possible for each person to use a different number of minutes and such a long list would be unreadable. A grouped frequency table, histogram, or frequency polygon would be most appropriate. c. You would present grouped data because time to complete carried out to seconds would produce too many unique numbers to organize meaningfully without groupings. A grouped frequency table, histogram, or frequency polygon would be most appropriate. d. You would present individual data values because number of siblings tends to take on limited values. A frequency table, histogram, or frequency polygon would be most appropriate. 2.39 INTERVAL FREQUENCY 18–20 2 15–17 6 12–14 2 9–11 3 6–8 7 3–5 8 2.40 The stem-and-leaf plot is depicted below: 3 3 2 0026 1 89 0 2 2.41 The stem-and-leaf plot is depicted below: 6 0 5 01367 4 002234489 3 13779 2 33567999 1 89 2.42 a. A histogram of grouped frequencies b. Approximately 32 c. Approximately 27 d. Two questions we might ask are (1) How close is the person to those photographed?, and (2)What might account for the two peaks in these data? e. 10 5 20 15 35 30 25 0 0 1 2 3 4 Number of television sets Percentage f. 2 4 6 8 0 1.5 4.5 0 7.5 10.5 13.5 16.5 19.5 Number of people pictured Frequency g. The data have two high points around 3–9 and 15–18. We can see that the data are asymmetric to the right, creating positive skew. h. The stem-and-leaf plot is depicted below: 2 0 1 01235566778 0 013333345567777889 Nolan3e_interior_Appendix_C_Ch02.indd 7 11/04/14 6:53 PM

  9. Page 9
    Read page 9 as text

    C-8 APPENDIX C i. These data reflect a floor effect because most of the observations are clustered on the lower end of the distribution between 0 and 9. This floor effect is likely caused by the fact that people cannot have fewer than 0 pictures of others. 2.43 a. MONTHS FREQUENCY PERCENTAGE 12 1 5 11 0 0 10 1 5 9 1 5 8 0 0 7 1 5 6 1 5 5 0 0 4 1 5 3 4 20 2 2 10 1 3 15 0 5 25 b. c. d. INTERVAL FREQUENCY 10–14 months 2 5–9 months 3 0–4 months 15 0 1 2 3 4 5 1 0 2 3 4 5 6 7 8 9 10 11 12 13 Frequency 2 3 4 5 1 6 0 2 1 0 3  1 4 5 6 7 8 9 10 11 12 13 Months Frequency e. 1 2 3 6 5 4 0 8 9 10 13 12 15 14 11 7 0 2.5 5 7.5 12.5 10 15 Months Frequency f. g. These data are centered around the 3-month period, with positive skew extending the data out to the 12-month period. h. The bulk of the data would need to be shifted from the 3-month period to approximately 12 months, so the women who have breast-fed for 3 months so far might be the focus of attention. Perhaps early contact at the hospital and at follow-up visits after birth would help encourage mothers to breast-feed, and to breast-feed longer. One could also consider studying the women who create the positive skew to learn what unique characteristics or knowledge they have that influenced their behavior. 2.44 a. The column for faculty shows a high point from 0–7 friends. b. The column for students shows two high points around 4–11 and 16–23, with some high outliers creating positive skew. c. The independent variable would be status, with two levels (faculty, student). d. The dependent variable would be number of friends. e. A confounding variable could be age, as faculty are older than students and tend to be less involved in social activities or situations where making friends is common. f. The dependent variable could be operationalized as the number of people who appear in photographs on display in dorm rooms and offices across campus, as was done for this 2 3 4 5 1 6 8 7 9 11 12 13 14 10 15 0 0 2.5  2.5 5 7.5 10 12.5 15 17.5 Months Frequency Nolan3e_interior_Appendix_C_Ch02.indd 8 11/04/14 6:53 PM

  10. Page 10
    Read page 10 as text

    APPENDIX C C-9 study. There are several additional ways these data could be operationalized. One way would be to record the number of Facebook friends each person has. Another way would be to count the number of friends each person reports interacting with on a regular basis. This latter method of measuring number of friends is more likely to reveal the quality of friendship via the amount of interaction. 2.45 a. FORMER STUDENTS NOW IN TOP JOBS FREQUENCY PERCENTAGE 13 1 1.85 12 0 0.00 11 0 0.00 10 0 0.00 9 1 1.85 8 3 5.56 7 4 7.41 6 5 9.26 5 9 16.67 4 8 14.81 3 23 42.59 b. c. d. This distribution is positively skewed. e. The researchers operationalized the variable of mentoring success as numbers of students placed into top professorial 5 10 15 20 25 0 3 0 4 1 2 5 6 7 8 10 11 12 13 9 Number of students mentored by each different professor Frequency 5 10 15 20 25 0 0 Number of students mentored by each different professor Frequency 1 2 3 4 5 6 7 8 9 10 11 12 14 16 15 13 positions. There are many other ways this variable could have been operationalized. For example, the researchers might have counted numbers of student publications while in graduate school or might have asked graduates to rate their satisfaction with their graduate mentoring experiences. f. The students might have attained their positions as professors because of the prestige of their advisor, not because of his mentoring. g. There are many possible answers to this question. For example, the attainment of a top professor position might be predicted by the prestige of the institution, the number of publications while in graduate school, or the graduate student’s academic ability. C H A P T E R 3 3.1 The five techniques for misleading with graphs are the biased scale lie, the sneaky sample lie, the interpolation lie, the extrapolation lie, and the inaccurate values lie. 3.2 (1) Organize the data by participant; each participant will have two scores, one on each scale variable. (2) Label the horizontal x -axis with the name of the independent variable and its possible values, starting with 0 if practical. (3) Label the vertical y -axis with the name of the dependent variable and its possible values, starting with 0 if practical. (4) Make a mark on the graph above each study participant’s score on the x -axis and across from his or her score on the y -axis. 3.3 To convert a scatterplot to a range-frame, simply erase the axes below the minimum score and above the maximum score. 3.4 A linear relation between variables means that the relation between variables is best described by a straight line. 3.5 With scale data, a scatterplot allows for a helpful visual analysis of the relation between two variables. If the data points appear to fall approximately along a straight line, the variables may have a linear relation. If the data form a line that changes direction along its path, the variables may have a nonlinear relation. If the data points show no particular relation, it is possible that the two variables are not related. 3.6 A line graph is used to illustrate the relation between two scale variables. One type of line graph is based on a scatterplot and allows us to construct a line of best fit that represents the predicted y scores for each x value. A second type of line graph allows us to visualize changes in the values on the y -axis over time. A time plot, or time series plot, is a specific type of line graph. It is a graph that plots a scale variable on the y -axis as it changes over an increment of time (e.g., second, day, century) recorded on the x -axis. 3.7 A bar graph is a visual depiction of data in which the independent variable is nominal or ordinal and the dependent variable is scale. Each bar typically represents the mean value of the dependent variable for each category. A Pareto chart is a specific type of bar graph in which the categories along the x- axis are ordered from highest bar on the left to lowest bar on the right. 3.8 Bar graphs typically depict summary statistics, such as frequencies or averages, for several different levels of one or more nominal or ordinal independent variables. Histograms typically depict frequencies for different values of one scale variable. Bars represent counts or percentages for different values of a scale variable or for different intervals of that scale variable. Nolan3e_interior_Appendix_C_Ch02.indd 9 11/04/14 6:53 PM

  11. Page 11
    Read page 11 as text

    C-10 APPENDIX C 3.9 A pictorial graph is a visual depiction of data typically used for a nominal independent variable with very few levels (categories) and a scale dependent variable. Each level uses a picture or symbol to represent its value on the scale dependent variable. A pie chart is a graph in the shape of a circle, with a slice for every level. The size of each slice represents the proportion (or percentage) of each category. In most cases, a bar graph is preferable to a pictorial graph or a pie chart. 3.10 Bar graphs are straightforward presentations of data, whereas the elements of pictorial graphs and pie charts can often distract from the data being presented. Also, mistakes in presentation style are much more common for pictorial graphs and pie charts than for bar graphs. 3.11 The independent variable typically goes on the horizontal x- axis and the dependent variable goes on the vertical y -axis. 3.12 Whenever possible, graph axes should start at 0, although sometimes it is not practical to start at 0. For example, when the data do not contain low values (and including 0 would minimize the depiction of the actual data), cut marks should be used to indicate axes that do not start at 0. 3.13 Moiré vibrations are any visual patterns that create a distracting impression of vibration and movement. A grid is a background pattern, almost like graph paper, on which the data representations, such as bars, are superimposed. Ducks are features of the data that have been dressed up to be something other than merely data. 3.14 Geographic information systems are particularly powerful for analyzing demographic patterns or demographic differences in a variable. Knowing how several variables change over geographic regions could lead researchers to detect important relations among variables. 3.15 Like a traditional scatterplot, the locations of the points on the bubble graph simultaneously represent the values that a single case (or country) has on two scale variables. The graph as a whole depicts the relation between these two variables. 3.16 Unlike a scatterplot, this bubble graph depicts two additional variables, for a total of four variables. The bubble graph allows the reader to evaluate the relations among multiple variables at the same time and can depict more complex relations than can the traditional scatterplot. 3.17 Total dollars donated per year is scale data. A time plot would nicely show how donations varied across years. 3.18 Sorting people into the categories of “alumni who donated money” and “alumni who did not donate money” creates nominal data. We would use a bar graph to depict the numbers of alumni who did and did not donate. 3.19 a. The independent variable is gender and the dependent variable is video game score. b. Nominal c. Scale d. The best graph for these data would be a bar graph because there is a nominal independent variable and a scale dependent variable. 3.20 Nonlinear, because the data change direction around 4.00 on the x -axis. 3.21 Linear, because the data could be fit with a line drawn from the upper-left to the lower-right corner of the graph. 3.22 These graphs are missing titles and axis labels. The axes are also missing 0 values. 3.23 a. Bar graph b. Line graph; more specifically, a time plot c. The y -axis should go down to 0. d. The lines in the background are grids, and the three- dimensional effect is a type of duck. e. 3.20%, 3.22%, 2.80% f. If the y -axis started at 0, all of the bars would appear to be about the same height. The differences would be minimized. 3.24 These data have a minimum value of 273 and a maximum value of 342. Because the minimum value is far from 0, it is not practical to have the axis start at 0, so cut marks would be used. (However, we would include the full range of data — 0 to 342 — if omitting some of these numbers would be misleading.) We might then include every 10th value, starting at 270: 270 280 290 300 310 320 330 340 350 3.25 The minimum value is 0.04 and the maximum is 0.36, so the axis could be labeled from 0.00 to 0.40. We might choose to mark every 0.05 value: 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 3.26 a. The highest life expectancy is 82 years. The fertility rate associated with the highest life expectancy is 0.96. b. Yes, this seems to be a linear relation, with the data fitting a line moving from the upper-left to the lower-right corner of the graph. As the fertility rate increases, the life expectancy at birth decreases. 3.27 The relation between physical health and positive emotions seems to be positive, with the data fitting a line moving from the lower-left to the upper-right corner of the graph. As positive emotions increase, self-reported physical health also tends to increase. 3.28 The relation between positive emotions and GDP seems to be positive, where countries with larger GDPs (represented by larger, yellower dots) are more concentrated in the upper-right corner of the graph and those with smaller GDPs (represented by smaller, darker red dots) are more concentrated in the middle to lower-left corner of the graph. As GDP increases, positive emotions tend to increase. 3.29 a. The independent variable is height and the dependent variable is attractiveness. Both are scale variables. b. The best graph for these data would be a scatterplot (which also might include a line of best fit if the relation is linear) because there are two scale variables. c. It would not be practical to start the axis at 0. With the data clustered from 58 to 71 inches, a 0 start to the axis would mean that a large portion of the graph would be empty. We would use cut marks to indicate that the axis did not include all values from 0 to 58. (However, we would include the full range of data — 0 to 71 — if omitting some of these numbers would be misleading.) 3.30 a. The independent variable is time (i.e., week) and the dependent variable is mean depression level. b. Both variables are scale. Nolan3e_interior_Appendix_C_Ch03.indd 10 11/04/14 7:03 PM

  12. Page 12
    Read page 12 as text

    APPENDIX C C-11 c. The best graph for these data would be a time plot because the social worker is tracking depression levels over a period of time (20 weeks). 3.31 a. The independent variable is country and the dependent variable is male suicide rate. b. Country is a nominal variable and suicide rate is a scale variable. c. The best graph for these data would be a bar graph or a Pareto chart. Because there are six categories or countries to list along the x -axis, it may be best to arrange them in order from highest to lowest using a Pareto chart. d. A time series plot could show year on the x -axis and suicide rate on the y -axis. Each country would be represented by a different color line. 3.32 a. b. For the most part, the points on the scatterplot do not seem to indicate any particular relation, whether linear or curvilinear. Low-mileage days (50 to 70 miles) have some low-climb and some high-climb days, and mid-mileage days (90 to 110 miles) have some low-climb and some high-climb days. Only the two very long mileage days (around 120 miles) have low climbs, perhaps indicating a tiny relation. c. The cyclists experience both the mileage and climbs as difficult and tend to notice days on which both are high. The organizers want to convince cyclists to sign up and pay the trip costs so they can make money; a promise that long mileage days won’t have big climbs helps them recruit cyclists. The staff have no vested interest either way. 3.33 a. 9000 1000 2000 3000 4000 5000 6000 7000 8000 0 60 50 80 90 70 130 120 110 100 Miles Climb Relation Between Cycling Daily Mileage and Cycling Daily Climb in Feet 12 10 8 6 4 2 0 5 0 15 20 Canada United Kingdom Germany Japan United States Italy France 10 30 25 Percentage with university degree GDP Relation Between Percentage with University Degree and GDP (in trillions of $US) b. The percentage of residents with a university degree appears to be related to GDP. As the percentage with a university degree increases, so does GDP. c. It is possible that an educated populace has the skills to make that country productive and profitable. Conversely, it is possible that a productive and profitable country has the money needed for the populace to be educated. 3.34 a. b. These data suggest that, although there are some fluctuations, a slight increase in organ donation seems to have taken place between 2001 and 2010. c. There are many possible answers to this question. You might, for instance, be interested in how characteristics of families or types of deaths distinguish between agreeing to and declining to donate. You might also be interested in what changes over time may have correlated with the slight increase in donation rates; for example, has there been an increase in public service announcements encouraging organ donation? 3.35 a. The independent variable is the academic institution. It is nominal; the levels are the 10 colleges. b. The dependent variable is alumni donation rate. It is a scale variable; the units are percentages, and the range of values is from 50.2 to 62.6. c. The defaults will differ, depending on which software is used. Here is one example. 70.00% 60.00% 50.00% 40.00% 30.00% 20.00% 10.00% 0.00% Princeton University (NJ) Thomas Aquinas College (CA) Carleton College (MN) Williams College (MA) Amherst College (MA) Centre College (KY) Middlebury College (VT) Davidson College (NC) College of the Holy Cross (MA) Bowdoin College (ME) d. The redesigns will differ, depending on which software is used. In this example, we added a clear title and labeled the y -axis (being sure that it reads from left to right). We also eliminated the unnecessary lines in the background and the decimal places of each number on the y -axis. 15 0 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 5 10 Donation rate per million population Year Organ Donation Rates per Million in Canada, 2001–2010 Nolan3e_interior_Appendix_C_Ch03.indd 11 11/04/14 7:03 PM

  13. Page 13
    Read page 13 as text

    C-12 APPENDIX C 0% 10% 20% 30% 40% 50% 60% 70% Princeton University (NJ) Thomas Aquinas College (CA) Carleton College (MN) Williams College (MA) Amherst College (MA) Centre College (KY) Middlebury College (VT) Davidson College (NC) College of the Holy Cross (MA) Bowdoin College (ME) Alumni donation rate (percent) Alumni Donation Rates for Top 10 Colleges and Universities e. There are many possible answers to this question. One might want to identify characteristics of alumni who donate, methods of soliciting donations that result in the best outcomes, or characteristics of universities that have the highest rates. f. Pictures could be used instead of bars. For example, dollar signs might be used to represent the donation rate for each college. g. If the dollar signs become wider as they get taller, as often happens with pictorial graphs, the overall size would be proportionally larger than the increase in donation rate it is meant to represent. A bar graph is not subject to this problem because graphmakers are not likely to make bars wider as they get taller. 3.36 a. A Pareto chart is organized from the highest bar to the lowest bar—in this case, from highest GDP per capita to lowest GDP per capita—whereas a bar graph might be organized with the countries listed alphabetically from left to right. b. The Pareto chart allows us to make comparisons more easily than does the bar graph. Moreover, we can very easily identify the countries with the highest and lowest GDP. 3.37 a. One independent variable is time frame; it has two levels: 1945–1950 and 1996–1998. The other independent variable is type of graduate program; it also has two levels: clinical psychology and experimental psychology. b. The dependent variable is percentage of graduates who had a mentor while in graduate school. c. 100 80 60 40 20 0 Clinical Experimental Type of psychology graduate program Percentage Percentage of Mentoring by Time Frame and Type of Psychology Graduate Program 1945–1950 1996–1998 d. These data suggest that clinical psychology graduate students were more likely to have been mentored if they were in school in the 1996–1998 time frame than if they were in school during the 1945–1950 time frame. There does not appear to be such a difference among experimental psychology students. e. This was not a true experiment. Students were not randomly assigned to time period or type of graduate program. f. A time series plot would be inappropriate with so few data points. It would suggest that we could interpolate between these data points. It would suggest a continual increase in the likelihood of being mentored among clinical psychology students, as well as a stable trend, albeit at a high level, among experimental psychology students. g. The story based on two time points might be falsely interpreted as a continual increase of mentoring rates for the clinical psychology students and a plateau for the experimental psychology students. The expanded data set suggests that the rates of mentoring have fluctuated over the years. Without the four time points, we might be seduced by interpolation into thinking that the two scores represent the end points of a linear trend. We cannot draw conclusions about time points for which we have no data—especially when we have only two points, but even when we have more points. 3.38 A pie chart would include the four percentages as slices, but it might be difficult to make comparisons, particularly between percentages similar in size (e.g., 9% and 5%). A bar graph allows for easier comparisons among categories than does a pie chart. 3.39 a. The details will differ, depending on the software used. Here is one example. Current Recent Former 0 10 20 30 40 50 60 70 80 90 100 Series 1 b. The default options that students choose to override will differ. For the bar graph on the next page, we (1) added a title, (2) labeled the x -axis, (3) labeled the y -axis, (4) rotated the y -axis label so that it reads from left to right, and (5) eliminated the unnecessary key. Nolan3e_interior_Appendix_C_Ch03.indd 12 11/04/14 7:03 PM

  14. Page 14
    Read page 14 as text

    APPENDIX C C-13 100 40 20 80 60 0 Current Recent Former Percentage Type of student Percentage Satisfied with Graduate Advisors Among Current Students, Recent Graduates, and Former Students Who Did Not Complete a PhD 3.40 a. The details will differ, depending on the software used. Here is one example. “World’s deepest” trash bin Regular trash bin 0 10 20 30 40 50 60 70 80 Series 1 b. The default options that students choose to override will differ. For the bar graph here, we (1) added a title, (2) labeled the x -axis, (3) labeled the y -axis, (4) rotated the y -axis, so that it reads from left to right, and (5) eliminated the unnecessary key. Kilograms of trash Type of trash bin Type of Trash Bin and Kilograms of Trash “World’s deepest” trash bin Regular trash bin 0 10 20 30 40 50 60 70 80 3.41 a. The graph is a scatterplot: individual points are identified for two scale variables—academic standing and “hotness.” b. The variables are academic standing and “hotness.” c. The graph could be redesigned to get rid of moiré vibrations, such as the colored background; and the grid (the background pattern of graph paper) and duck (the woman in the background image) could be eliminated. 3.42 The examples will differ for each student. Correct answers will include the following types of variables. a. Frequency polygon: one scale variable; for example, on the x -axis, times for rats to complete a maze, and on the y -axis, frequencies for each time b. Line graph (line of best fit): two scale variables; for example, on the x -axis, hours of maze-training for rats, and on the y -axis, predicted times for rats to complete a maze c. Bar graph (one independent variable): one nominal or ordinal independent variable, such as gender of rat, on the x -axis, and one scale dependent variable, such as time to complete a maze, on the y -axis d. Scatterplot: two scale variables; for example, on the x -axis, hours of maze-training for rats, and on the y -axis, times for rats to complete a maze e. Time series plot: one time-related independent variable, such as year, on the x -axis, and one scale dependent variable, such as mean GPA of incoming students, on the y -axis f. Pie chart: Trick question! Don’t use one; use a bar graph instead. g. Bar graph (two independent variables): two nominal or ordinal independent variables, such as gender of rat and reinforcement schedule for rat, on the x -axis, and a scale dependent variable, such as time to complete a maze, on the y -axis 3.43 Each student’s advice will differ. The following are examples of advice. a. Business and women: Eliminate all the pictures, including the woman, piggy banks, the dollar signs in the background, and the icons to the right (e.g., house). The two bars near the top could mislead us into thinking they indicated quantity, even though they are the same length for two different median wages. Either eliminate the bars or size them so that they are appropriate to the dollars they represent. Ideally, the two median wages would be presented in a bar graph. Eliminate unnecessary words (e.g., “The Mothers of Business Invention”). b. Workforce participation: Eliminate all the pictures. A falling line in the art shown indicates an increase in percentage; notice that 40% is at the top and 80% is at the bottom. Make the y -axis go from highest to lowest, starting from 0. Make the lines easier to compare by eliminating the three- dimensional effect. Make it clear where the data point for each year falls by including a tick mark for each number on the x -axis. 3.44 The articles and subsequent responses will be different for each student. 3.45 a. The graph proposes that Type I regrets of action are initially intense but decline over the years, while Type II regrets of inaction are initially mild but become more intense over the years. Nolan3e_interior_Appendix_C_Ch03.indd 13 11/04/14 7:03 PM

  15. Page 15
    Read page 15 as text

    C-14 APPENDIX C b. There are two independent variables: type of regret (a nominal variable) and age (a scale variable).There is one dependent variable: intensity of regrets (also a scale variable). c. This is a graph of a theory. No data have been collected, so there are no statistics of any kind. d. The story that this theoretical relation suggests is that regrets over things a person has done are intense shortly after the actual behavior but decline over the years. In contrast, regrets over things a person has not done but wishes they had are initially low in intensity but become more intense as the years go by. 3.46 a. This is a time plot. The researchers chose this type of graph because they wanted to show changes in the number of psychology degrees over time. b. This graph suggests a fairly large increase in bachelor’s degrees over time, with smaller increases in master’s degrees and doctoral degrees. c. There are two independent variables. One is type of degree, with three levels: bachelor’s, master’s, and doctoral. It could be considered ordinal. The other independent variable is year; it is a scale variable. The dependent variable is number of psychology degrees; it is also a scale variable. d. There are several possible answers to this question. For example, the y -axis starts at 0, there is a clear title, and all labels read from left to right. e. There are several possible answers to this question. For example, the graph creator should have labeled the x -axis. The y -axis is too “busy”; intervals of 10,000 would be better. f. There are several possible answers to this question. For example, we could track percentage out of all such degrees (e.g., percentage of psychology bachelor’s degrees conferred out of all bachelor’s degrees conferred). g. There are several possible answers to this question. For example, we might examine what types of careers psychology undergraduates pursue that do not require a master’s or doctoral degree in psychology. 3.47 a. When first starting therapy, the client showed a decline, as measured by the Mental Health Index (MHI). After 8 weeks of therapy, this trajectory reversed and there was a week-to- week improvement in the client’s MHI. b. There are many possible answers. For example, the initial decline in the client’s MHI may have been due to difficulties in adapting to therapy that the client overcame while working with the therapist. Alternatively, it may be that the client initially entered therapy because of difficult life circumstances that continued through the first weeks of therapy but resolved after several weeks. c. Because the client is not beneath the failure boundary, and because the client experienced improvement over the last few weeks of therapy, it may be beneficial for the client to continue in therapy. 3.48 a. Density of traffic is represented by the thickness of the colored lines across the roads. The flow of traffic is represented by the color of the lines. b. Answers to this question will vary, depending on the time of day and the exact traffic conditions. c. This interactive graph allows anyone to see up-to-the- minute local, regional, and national traffic conditions. Traditional graphing techniques do not allow such up-to- date information on demand. 3.49 a. Data can almost always be presented more clearly in a bar graph or table than in a pie chart. b. Answers to this question should include revising the data to add up to 100%, removing chartjunk (e.g., colors, shading, background images, etc.), and more clearly labeling categories with candidate names only. The graph also should not have 3-D features. 3.50 a. This is a bar graph. b. Variables are boat (levels: Titanic versus other ships), passenger type (levels: women, children, men, crew, captain), survival rate, and death rate. c. No, both survival rate and death rate are not needed, as these two numbers add up to 100% for each passenger type. Both rates could have been included to either provide all relevant data, or to overemphasize the death rates in comparison to the survival rates. d. This graph is visualizing the different death and survival rates among various passenger types on the Titanic versus other ships in the study. Though women and children had higher survival rates, and lower death rates, than men, crew, and captain on the Titanic , the opposite is true for the other 16 ships in the study. Thus, it seems that the Titanic was not the norm for passenger survival of shipwrecks, and it was not typical for women and children to be saved first. 3.51 a. The independent variable is song type, with two levels: romantic song and nonromantic song. b. The dependent variable is dating behavior. c. This is a between-groups study because each participant is exposed to only one level or condition of the independent variable. d. Dating behavior was operationalized by giving one’s phone number to an attractive person of the opposite sex. This may not be a valid measure of dating behavior, as we do not know if the participant actually intended to go on a date with the researcher. Giving one’s phone number might not necessarily indicate an intention to date. e. We would use a bar graph because there is one nominal independent variable and one scale dependent variable. f. The default graph will differ, depending on which software is used. Here is one example: Romantic song Nonromantic song 0 10 20 30 40 50 60% Nolan3e_interior_Appendix_C_Ch03.indd 14 11/04/14 7:03 PM

  16. Page 16
    Read page 16 as text

    APPENDIX C C-15 g. The default options that students choose to override will differ. Here is one example. 3.52 a. Sunday at midnight b. Saturday at 9 a . m . c. Are people happiest on Saturdays? The independent variable is day of the week. The dependent variables are positive attitude and negative attitude. d. Mood is operationalized as positive or negative attitudes expressed in tweets sent via Twitter. This may not be a valid measure of mood, as an expression of one’s attitude may not reflect someone’s actual mood states. e. This finding that people have the highest average negative mood on Sunday nights fits with the hypothesis that people are happiest on Saturdays. Because Sunday night is the night before work or school resumes for many people, we might study whether the imminent start to the work and school week is affecting people’s moods. C H A P T E R 4 4.1 The mean is the arithmetic average of a group of scores; it is calculated by summing all the scores and dividing by the total number of scores. The median is the middle score of all the scores when a group of scores is arranged in ascending order. If there is no single middle score, the median is the mean of the two middle scores. The mode is the most common score of all the scores in a group of scores. 4.2 The mean can be estimated by examining a visual display of data, such as a histogram, and finding the point in the data that seems to create balance between the two sides of the distribution. The mean can be calculated precisely using arithmetic. In this case, we take all of the data points, sum their values, and then divide by the total number of scores. 4.3 The mean takes into account the actual numeric value of each score. The mean is the mathematic center of the data. It is the center balance point in the data, such that the sum of the deviations (rather than the number of deviations) below the mean equals the sum of deviations above the mean. 4.4 Unimodal distributions have one mode. Bimodal distributions have two modes. Multimodal distributions have more than two modes. Romantic song Nonromantic song 0 10 20 30 40 50 60 Percentage of Women Who Gave Phone Number Based on Song Type Percentage who gave phone number Song type 4.5 The mean might not be useful in a bimodal or multimodal distribution because in a bimodal or multimodal distribution the mathematical center of the distribution is not the number that describes what is typical or most representative of that distribution. 4.6 An outlier is an extreme score that is either very high or very low in comparison with the rest of the scores in a sample. 4.7 The mean is affected by outliers because the numeric value of the outlier is used in the computation of the mean. The median typically is not affected by outliers because its computation is based on the data in the middle of the distribution, and outliers lie at the extremes of the distribution. 4.8 The mode is typically used in one of three situations: (1) when one score dominates the distribution, (2) to describe bimodal or multimodal distributions, and (3) when nominal data are summarized. 4.9 The standard deviation is the typical amount each score in a distribution varies from the mean of the distribution. 4.10 SD 2 comes from the words standard deviation squared. X represents the sample scores, M the sample mean, and N the number of scores in the sample. The o indicates that values need to be summed. 4.11 The standard deviation is a measure of variability in terms of the values of the measure used to assess the variable, whereas the variance is squared values. Squared values simply don’t make intuitive sense to us, so we take the square root of the variance and report this value, the standard deviation. 4.12 a. 1. m ; SD 2 2. The symbol for the mean should be capitalized, and the second symbol is for variance, not for standard deviation. 3. M ; SD b. 1. μ 2. Greek letters are used for parameters; this is a statistic because it’s from a sample. 3. M c. 1. X highest 2 X lowest 2. The X ’s should be italicized. 3. X highest 2 X lowest 4.13 The range is the difference between the highest score and the lowest score in the data set. Thus, the range is completely driven by the most extreme scores in the data set and is susceptible to the effects of outliers. The interquartile range is based on the middle 50% of the data. Unlike the range, it is not affected by the effects of outliers. 4.14 The median is the score that splits an ordered distribution of numbers in half. The first quartile of a data set is simply the median of the first half of that data set. Similarly, the third quartile of a data set is simply the median of the second half of that data set. 4.15 The first quartile is the 25th percentile. 4.16 The third quartile is the 75th percentile. Nolan3e_interior_Appendix_C_Ch03.indd 15 11/04/14 7:03 PM

124 more pages in the full document

Unlock it once and it stays in your library, ready to chat with or turn into flashcards.

Unlock the full document

More Statistics study materials

See all

Study this document with CramX

Once it is in your library, every tool can work from it.