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Essential Statistics for the Behavioral Sciences Brief Edition Solution Manual

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    1 Problem Solutions CHAPTER 1 1. Descriptive statistics are procedures used to summarize, organize, and make sense of a set of scores or observations, whereas inferen- tial statistics are procedures that allow researchers to infer or gener- alize observations made with samples to the larger population from which they were selected. 2. Data describe a set of measurements (made up of raw scores); a raw score describes individual measurements. 3. Samples are selected from populations of interest. Hence, samples consist of a portion of individuals in a population of interest. 4. Experimental, quasi-experimental, and correlational research methods. 5. (a) Independent variable. (b) Dependent variable. 6. The four scales of measurement are nominal, ordinal, interval, and ratio. Ratio scale measurements are the most informative. 7. Yes. Nominal values are often coded (i.e., converted to numeric values) when entered into statistical software. 8. Interval variables do not have a true zero, and ratio variables do have a true zero. 9. Amount; Class. 10. Continuous and discrete variables. 11. (a) Descriptive statistics. (b) Inferential statistics. (c) Descriptive statistics. 12. (a) False. (b) True. (c) True. 13. No, it is not necessary to make inferences, because all individuals in the population were observed. 14. The statistics class has a population of 25 students enrolled, but a sample of only 23 students attended. 15. By definition, samples will always be smaller than the population. Samples constitute a subset (or a smaller set of values) than those in the population. 16. An experimental research method because the researcher claims to have demonstrated cause. 17. A correlational research method because pairs of scores (height, income) were measured and compared for each participant. 18. (a) Quasi-independent variable. (b) Quasi-independent variable. (c) Independent variable. (d) Independent variable. (e) Quasi- independent variable. (f) Independent variable. 19. (a) Sleeping pill (real or fake). (b) Time spent sleeping. •••

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    2 Statistics for the Behavioral Sciences 20. (a) Cocaine use (dependent vs. inexperienced). (b) Impulsive behavior. 21. (a) Nominal. (b) Yes, it is appropriate to numerically code “months” because it is a nominal scale variable. 22. Nominal, ordinal, interval, and ratio. 23. The main disadvantage of measuring qualitative data is that the data are on a nominal scale, which limits the types of conclusions that researchers can draw. For this reason, quantitative variables are more often measured because they are more informative in terms of order, differences, and ratios. 24. (a) Qualitative. (b) Quantitative. (c) Quantitative. (d) Qualitative. 25. (a) Continuous. (b) Discrete. (c) Continuous. (d) Discrete. 26. Variable Qualitative vs. Quantitative Continuous vs. Discrete Type of Measurement Sex Qualitative Discrete Nominal Seasons Qualitative Discrete Nominal Time of day Quantitative Continuous Ratio Rating scale score Quantitative Discrete Interval Movie ratings (1 to 4 stars) Quantitative Discrete Ordinal Number of students in your class Quantitative Discrete Ratio Temperature (degrees Fahrenheit) Quantitative Continuous Interval Time (in minutes) to prepare dinner Quantitative Continuous Ratio Position standing in line Quantitative Discrete Ordinal 27. (a) Descriptive statistics. (b) Overall, gun ownership really has not changed in the last 40 years with the percent of gun owners in 1972 being identical to the percent of gun owners in 2012. 28. An operational definition. 29. No, an experimental method is not possible because the variable (sex) is a preexisting variable; it is a quasi-independent variable. 30. (a) Continuous. (b) Quantitative. (c) Ratio scale. 31. The 91,373 nineteen-year-old-men likely represent a sample of all 19-year-old men in a much larger population. 32. Equidistant scales. Variables on an interval scale have equidistant scales, meaning that differences on this scale are informative.

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    Problem Solutions 3 CHAPTER 2 1. Step 1: Find the real range. Step 2: Find the interval width. Step 3: Construct the frequency distribution. 2. Grouped data are distributed in intervals; ungrouped data are not. 3. (a) No. (b) Yes. 4. To ensure that a single score cannot be counted in more than one interval. 5. A percentile rank. 6. Ungrouped data sets with only a few different scores, and qualitative or categorical variables. 7. Rule 1: A vertical rectangle represents each interval, and the height of the rectangle equals the frequency recorded for each interval. Rule 2: The base of each rectangle begins and ends at the upper and lower boundaries of each interval. Rule 3: Each rectangle touches adjacent rectangles at the boundaries of each interval. 8. Midpoint; Upper boundary. 9. When the data are discrete. Histograms are only used with continuous data. 10. Discrete/categorical data. 11. (a) Intervals f(x) 18–22 5 13–17 3 8–12 7 3–7 5 (b) Interval: 8–12. 12. (a) Classes f(x) L 9 C 16 R 5 (b) Yes, the rat did press the center lever the most. 13. The intervals overlap at the upper and lower boundaries, which might lead to some scores being counted in more than one interval. 14. Three errors are (1) the intervals overlap, (2) the class width for each interval is not equal, and (3) the distribution includes an open interval.

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    4 Statistics for the Behavioral Sciences 15. The lower boundaries are 35, 46, 57, and 68. 16. The upper boundaries are 3, 6, 9, 12, 15, and 18. 17. No, the data should remain ungrouped because the data are categorical. 18. The interval width for each interval is 3. 19. (a) Number of Dreams Percentile Rank 4 100% 3 88% 2 60% 1 24% 0 10% (b) Two Dreams 20. Sixty children qualify for the new cognitive-behavioral therapy. 21. (a) A bar chart or pie chart if distributing the letter grades along the x -axis. A histogram or frequency polygon if distributing the numeric values on a 4.0 grading scale. (b) A bar chart or pie chart. (c) A bar chart or pie chart distributing the frequencies for each behavioral ther- apy. (d) A histogram or frequency polygon. 22. (a) Histogram. (b) Bar chart. (c) Histogram. (d) Bar chart. 23. A = 8, B = 3, C = 12 24. (a) A = 78, B = 86, C = 68, D = 13. (b) Yes, this was a difficult test because half the class would fail. 25. (a) 35 students. (b) 6 students. (c) 15 students. 26. 100% Cumulative Percent Intervals 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% 2 4 6 8 10 12

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    Problem Solutions 5 27. (a) 0 5 10 15 20 25 30 0 Frequency Intervals 8 7 6 5 4 3 2 1 (b) 10 Frequency Season of Birth 9 8 7 6 5 4 3 2 1 0 Spring Summer Fall Winter Spring Summer Fall Winter 28.

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    6 Statistics for the Behavioral Sciences 29. The interval of 60–79 has the largest portion of students (20% of stu- dents fall in this interval). 30. The percentile point for the 50th percentile is 74.5 minutes. 31. (a) Real range = 56. (b) 30%. 32. (a) While more men earned a bachelor’s degree in psychology in 1970– 1971, women earn more than three times the number of bachelor’s degrees in psychology as of 2005–2006. (b) Ungrouped data, because years are not distributed consecutively. 33. (a) A relative percent distribution. (b) About 576 Americans. CHAPTER 3 1. (a) N . (b) n . (c) μ . (d) M or X . 2. Measures of central tendency are statistical measures used to locate a single score that is most representative or descriptive of all scores in a distribution. 3. The median is the score in the middle of a distribution. It is always at the center of a distribution. 4. A population mean is the mean for a set of scores in an entire popula- tion, whereas the sample mean is the mean for a sample, or subset of scores from a population. 5. Five characteristics of the mean are as follows: (1) Changing an exist- ing score will change the mean; (2) adding a new score or completely removing an existing score will change the mean, unless that value equals the mean; (3) adding, subtracting, multiplying, or dividing each score in a distribution by a constant will cause the mean to change by that constant; (4) the sum of the differences of scores from their mean is zero; and (5) the sum of the squared differences of scores from their mean is minimal. 6. The weighted mean equals the arithmetic mean when the sample sizes or “weights” for a set of scores are the same or equal. 7. Data that are normally distributed on an interval or ratio scale of mea- surement. 8. Data that are skewed and ordinal data. 9. The mode is used with other measures of central tendency for any modal distribution and for nominal data. 10. (a) Median. (b) Mean. (c) Mean. 11. Mean = 4, median = 2, mode = 0. 12. (a) College students: mean = 25, median = 18, mode = 21. Parents: mean = 14, median = 18, mode = 21. (b) Because both distributions are skewed, the median would be the appropriate measure of central ten- dency. This might be misleading, though, because the median indicates that texting was the same between groups (the median was 18 in both samples), even though differences exist in regard to the mean.

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    Problem Solutions 7 13. The mean because the data are distributed normally. 14. The mean because the data are normally distributed, and the duration (in hours) is a ratio scale measure. 15. (a) Positively skewed distribution. (b) The median. 16. The median because the data are negatively skewed. 17. The mode because the data are nominal. 18. Bimodal distribution. 19. Weighted mean = 14.69. 20. Weighted mean = 3.65. 21. (a) The mean, because time is a ratio measurement. (b) The mode, because blood types are nominal data. (c) The median, because college rankings are ordinal data. 22. (a) M = 22 points. (b) M = 10 points. (c) M = 24 points. (d) M = 6 points. 23. M = 6 new friends. 24. The sum of the differences of scores from the mean is 0. 25. The mean will decrease. 26. The mean will increase. 27. (a) The mean will increase. (b) The mean will not change. (c) The mean will decrease. 28. The new mean weight is 190 pounds. 29. Yes, the mean would be appropriate because the data are approximately normally distributed (the mean and median do not differ greatly). 30. (a) Heavy-drinking men have higher mean self-esteem scores than heavy-drinking women. 31. (a) The mode for nonsmoking Black participants was concern about a family member; the mode for nonsmoking White participants was concern for a friend. 32. Yes, participants consumed the most candies in the proximate and vis- ible condition. 33. The median because these data are negatively skewed. CHAPTER 4 1. No. A data set can vary (variability is greater than 0) or not vary (vari- ability is equal to 0). A negative variability is meaningless. 2. Two scores; the largest and smallest score in a distribution. 3. The IQR is the range of data after the top and bottom 25% of scores are removed. 4. The variance is preferred because it includes all scores to estimate vari- ability.

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    8 Statistics for the Behavioral Sciences 5. Deviations are squared for three reasons: (1) The sum of the differences of scores from their mean will always equal 0. To avoid having 0 in the numerator, each deviation is squared. (2) The sum of the squared dif- ferences of scores from their mean is minimal. Hence, the solution we obtain is the smallest possible positive value—or the value with min- imal error. (3) Squaring scores can be corrected by taking the square root of the variance. 6. The variance of the sample will equal the variance of the population from which the sample was selected, on average. 7. ( n − 1). 8. The standard deviation measures the average distance that scores devi- ate from their mean. 9. The empirical rule states that 68% of all scores lie within 1 SD of the mean, 95% of all scores lie within 2 SD of the mean, and 99.7% of all scores lie within 3 SD of the mean. 10. The standard deviation is always positive; is used to describe quantita- tive variables, typically reported with the mean; and is affected by the value of every score in a distribution. 11. (a) R = 98 − 24 = 74. (b) IQR = 94 − 77 = 17. (c) SIQR = 8.5. (d) σ 2 = 448.64. (e) σ = 21.18. 12. (a) R = 98 − 77 = 21. (b) IQR = 96 − 85 = 11. (c) SIQR = 5.5. (d) s 2 = 55.41. (e) SD = 7.44. 13. Yes, Sample 2 has a larger range because the data deviate further from their mean. 14. (a) df = 30. (b) s 2 120 30 = = 4.0, SD = 4 = 2. 15. (a) df = 1,200. (b) s 2 10 800 1 200 = , , = 9.0, SD = 9 = 3. 16. No, in both cases, the standard deviation is the square root of the vari- ance. Hence, the population and the sample standard deviation will be 9 3 0 = . . 17. No, in both cases, the variance is the square of the standard deviation. So the population and sample variance will be 12 2 = 144. 18. (a) Decrease. (b) Increase. (c) No effect. 19. (a) Increase. (b) No effect. (c) Decrease. 20. 36. 21. 6. 22. s SD 2 240 60 1 4 07 4 07 2 02 = = = = − . ; . . . 23. Yes, we would divide by N : σ 2 = 240/60 = 4; σ = 4 = 2.00. 24. (a) SD = 4. (b) SD = 8. 25. SS = 69.60, s 2 = 7.73, SD = 2.78. 26. SS = 37.50, s 2 = 5.36, SD = 2.31. 27. (a) Population: SS = 1,755, σ 2 = 109.69, σ = 10.47. (b) Sample: SS = 1,755, s 2 = 117.00, SD = 10.82.

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    Problem Solutions 9 28. SD = 0.5. 29. (a) df = 60 for men; df = 92 for women. (b) Men are more likely to smoke at least half a pack per day because 10 cigarettes is within 1 SD of the mean for men but greater than 1 SD from the mean for women. 30. (a) Yes, husbands and wives show a similar distribution in their ratings of love. (b) The ratings of love data are negatively skewed for husbands, and for wives. Thus, most husbands, and most wives, appear to give higher ratings of love, with a few giving much lower ratings. 31. No, the data are likely skewed because the mean is pulled toward the larger scores in the distribution. 32. Men showed greater variability overall because the standard deviations for men were larger than for women. CHAPTER 5 1. Probability ( p ( x )) is calculated by dividing the number of times an out- come occurs ( f ( x )) by the total number of possible outcomes (sample space). 2. Two characteristics of probability: Probability varies between 0 and 1 and can never be negative. 3. The normal distribution is a theoretical distribution with data that are symmetrically distributed around the mean, median, and mode. 4. Normality arises naturally in many physical, behavioral, and social measurement studies. 5. The eight characteristics are as follows: (1) The normal distribution is mathematically defined. (2) The normal distribution is theoretical. (3) The mean, median, and mode are all located at the 50th percentile. (4) The normal distribution is symmetrical. (5) The mean can equal any value. (6) The standard deviation can equal any positive value. (7) The total area under the curve of a normal distribution is equal to 1.00. (8) The tails of a normal distribution are asymptotic. 6. The mean equals 0, and the standard deviation equals 1. 7. A z score is a unit of measurement distributed along the x -axis of a standard normal distribution. Each z score reflects a standard deviation along the x -axis. 8. The standard normal transformation is the difference between a score and the mean, divided by the standard deviation. 9. Step 1: Transform a raw score ( x ) into a z score. Step 2: Locate the corresponding proportion for the z score in the unit normal table. 10. Step 1: Locate the z score associated with a given proportion in the unit normal table. Step 2: Transform the z score into a raw score ( x ). 11. (a) p = .375. (b) p = .625. (c) p = 0.

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    10 Statistics for the Behavioral Sciences 12. 3,000 mothers were not satisfied. 13. (a) .1587. (b) .8531. (c) .5000. (d) .9974. (e) .0250. 14. (a) .6915. (b) .0934. (c) .5000. (d) .0250. (e) .4602. 15. (a) .4750. (b) .0000. (c) .8664. (d) .0781. (e) .1359. 16. (a) The areas are equal. (b) The first area is bigger. (c) The first area is bigger. (d) The second area is bigger. (e) The second area is bigger. 17. p = .8413, Hence, 84.13% of players on the team can run the 100- meter dash in faster than 0.14 second. 18. (a) z = 1.645. (b) z = −1.96. (c) z = −0.51. (d) z = 0. (e) z = 0. 19. (a) p = .6826. (b) x = 26.4. (c) .0808. (d) .9192. 20. Students with a 3.56 GPA or higher will be offered a scholarship. 21. (a) The areas are equal. (b) The first area is bigger. (c) The second area is bigger. (d) The areas are equal. (e) The second area is bigger. 22. (a) .0548. (b) .1587. (c) .4452. (d) .8413. (e) .9452. 23. SD = 3. 24. SD = 1.50. 25. M = 136. 26. M = 10. 27. She knows that very few people will be highly accurate at lie detec- tion because the data are normally distributed—behaviors far from the mean (e.g., highly accurate lie detection) will be associated with a low probability of occurrence. 28. (a) z = 0. (b) z = 0. (c) The z scores are the same because both scores are at the mean in their respective distributions. 29. (a) p = .9049. (b) p = .0418. 30. (a) p = .1587. (b) p = .0122. 31. (a) p = .7764. 32. (a) Fast group. (b) Slow group. CHAPTER 6 1. The sampling distribution is a distribution of sample statistics that could be obtained from all possible samples of a given size from a pop- ulation. 2. Sampling without replacement and conditional probabilities are related in that when we sample without replacement, the probability of each selection is conditional or dependent on the person or item that was selected in the previous selection. 3. When order matters, each time we select participants in a different order, it is counted as a different possible sample. When order does not matter, selecting participants in a different order is counted as the same sample.

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