Solution Manual for Essentials of Statistics for the Behavioral Sciences, 2nd Edition

APPENDIX C C-3 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. e. We could have recruited a sample of people who were HIV-positive. Half would have been randomly assigned to take the oral vaccine; half would have been randomly assigned to take something that appeared to be an oral vaccine but did not have the active ingredient. They would have been followed to determine whether they developed cholera. tracking weight loss leads participants to implement weightloss 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 noexercise 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. 1.31 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. 1.32 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. 1.36 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 a period of time to measure change during that time 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. 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. 1.33 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.” 1.34 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 cohabitating 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 cohabitate. This variable could be operationalized and measured via household income. 1.35 a. Researchers could have randomly assigned some people who are HIV-positive to take the oral vaccine and other CHAPTER 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 C-4 APPENDIX C 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 2.5 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. 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. 2.19 0.04, 198.22, and 17.89 2.20 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. 2.21 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. 2.22 37.5, 52.5, and 67.5 2.23 25 shows 2.24 Twelve countries had between 2 and 10 first- or second-place World Cup finishes. 2.6 The x-axis is typically labeled with the name of the variable of interest. The y-axis is typically labeled “Frequency.” 2.25 Serial killers would create positive skew, adding high numbers 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. 2.26 People convicted of murder are assumed to have killed at least 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. of murders to the data that are clustered around 1. one person, so observations below one are not seen, which creates a floor effect. 2.27 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. 2.28 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. 2.29 a. PERCENTAGE FREQUENCY PERCENTAGE 2.11 With positively skewed data, the distribution’s tail extends to 10 1 5.26 the right, in a positive direction, and with negatively skewed data, the distribution’s tail extends to the left, in a negative direction. 9 0 0.00 8 0 0.00 7 0 0.00 6 0 0.00 5 2 10.53 2.13 A ceiling effect occurs when there are no scores above a 4 2 10.53 certain value; a ceiling effect leads to a negatively skewed distribution because the upper part of the distribution is constrained. 3 4 21.05 2 4 21.05 1 5 26.32 0 1 5.26 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. 2.14 4.98% and 2.27% 2.15 17.95% and 40.67% 2.16 3.69% and 18.11% are scale variables, both as counts and as percentages. 2.17 0.10% and 96.77% 2.18 1,889.00, 2.65, and 0.08 b. 10.53% of these schools had exactly 4% of their students report 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. APPENDIX C C-5 d. f. Eight 9 2.31 a. The variable of alumni giving was operationalized by the 8 percentage of alumni who donated to a given school. There are several other ways it could be operationalized. For example, the data might consist of the total dollar amount or the mean dollar amount that each school received. 7 6 Frequency b. 5 INTERVAL FREQUENCY 4 60–69 1 3 50–59 0 2 1 0 0 1 3 5 7 9 Percent of students 11 e. One f. The data are clustered around 1% to 4%, with a high outlier, 10%. 2.30 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. e. 12 11 10 9 8 7 Frequency 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Years 40–49 6 30–39 15 20–29 21 10–19 24 0–9 3 c. There are many possible answers to this question. For example, we might ask whether sports team success predicts alumni giving or whether the prestige of the institution is a factor (the higher the ranking, the more alumni who donate). d. 25 20 15 Frequency 10 5 0 0 5 15 25 35 45 55 65 Percentage of alumni donations e. 25 20 15 Frequency 10 5 0 25 5 15 25 35 45 55 65 75 Percentage of alumni donations f. There is one unusual score—61. The distribution appears to be positively skewed. The center of the distribution seems to be in the 10–29 range. C-6 APPENDIX C 2.32 a. INTERVAL FREQUENCY 60–69 1 50–59 7 40–49 10 30–39 7 20–29 2 10–19 3 a long list would be unreadable. Grouped frequency table, histogram, or frequency polygon 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. Grouped frequency table, histogram, or frequency polygon d. You would present individual data values because number of siblings tends to take on limited values. Frequency table, histogram, or frequency polygon 2.35 a. b. 10 9 8 INTERVAL FREQUENCY 300–339 4 260–299 7 220–259 9 180–219 3 7 b. This is not a random sample because only résumés from those applying for a receptionist position in his office were included in the sample. c. This information lets the trainees know that most of these résumés contained between 220 and 299 words. This analysis tells us nothing about how word count might relate to quality of résumé. 6 Frequency 5 4 3 2 1 2.36 a. A histogram of grouped frequencies 0 5 15 25 35 45 55 65 Wins c. The summary will differ for each student but should include the following information: the data appear to be roughly symmetric, maybe a bit negatively skewed. d. There are many possible answers to this question. For example, one might ask whether teams with older players do better or worse than those with younger players. Another study might examine whether team budget relates to wins; there’s a salary cap, but some teams might choose to pay the “luxury tax” in order to spend more. Does spending make a difference? 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. INTERVAL FREQUENCY 2.33 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. f. 18–20 2 15–17 6 12–14 2 9–11 3 6–8 7 3–5 8 8 6 Frequency 4 2 0 0 1.5 4.5 7.5 10.5 13.5 16.5 19.5 Number of people pictured 2.34 a. You would present individual data values because the few categories of eye color would result in a readable list. Frequency table b. You would present grouped data because it is possible for each person to use a different amount of minutes and such 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. APPENDIX C C-7 2.37 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 5 4 3 Frequency 2 1 c. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 4 Frequency 3 2 2.5 5 7.5 10 12.5 15 Months f. 15 0 2.5 5 7.5 10 12.5 15 17.5 Months 2.38 a. The column for faculty shows a high point from 0–7 1 0 0 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 3month period to approximately 12 months, so that group of women 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 breastfeed 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. 6 5 d. 15 14 13 12 11 10 9 Frequency 8 7 6 5 4 3 2 1 0 14 13 12 11 10 9 Frequency 8 7 6 5 4 3 2 1 0 22.5 b. 0 e. 21 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Months INTERVAL FREQUENCY 10–14 months 2 5–9 months 3 0–4 months 15 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 C-8 APPENDIX C d. The researchers operationalized the variable of mentoring success as numbers of students placed into top professorial 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. e. The students might have attained their professor positions because of the prestige of their advisor, not because of his mentoring. f. 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. in dorm rooms and offices across campus, as was done for this 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. 2.39 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 CHAPTER 3 8 3 5.56 3.1 7 4 7.41 The biased scale lie, the sneaky sample lie, the interpolation lie, the extrapolation lie, and the inaccurate values lie. 6 5 9.26 3.2 5 9 16.67 4 8 14.81 3 23 42.59 (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. 15 3.4 A linear relation between variables means that the relation between variables is best described by a straight line. 10 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, this indicates a linear relation. If the data form a line that changes direction along its path, a nonlinear relation may be present. 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 xaxis are ordered from highest bar on the left to lowest bar on the right. a. 25 20 Frequency 5 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Number of students b. 25 20 15 Frequency 10 5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Number of students mentored by each different professor c. This distribution is positively skewed.