Solution Manual for Biostatistics for the Biological and Health Sciences, 2nd Edition

Section 2-1: Frequency Distributions for Organizing and Summarizing Data 7 Chapter 2: Exploring Data with Tables and Graphs Section 2-1: Frequency Distributions for Organizing and Summarizing Data 1. The table summarizes measurements from 40 subjects. It is not possible to identify the exact values of all of the original cotinine measurements. 2. The classes of 0–100, 100–200, …, 400–500 overlap, so it is not always clear which class we should put a value in. For example, the value of 100 could go in the first class or the second class. The classes should be mutually exclusive. 3. Cotinine (ng/Ml) 0–99 100–199 200–299 300–399 400–499 Relative Frequency 27.5% 30.0% 35.0% 2.5% 5.0% 4. The sum of the relative frequencies is 125%, but it should be 100%, with a small round off error. All of the relative frequencies appear to be roughly the same, but if they are from a normal distribution, they should start low, reach a maximum, and then decrease. 5. Class width: 100 Class midpoints: 49.5, 149.5, 249.5, 349.5, 449.5, 549.5 Class boundaries: –0.5, 99.5, 199.5, 299.5, 399.5, 499.5, 599.5 6. Class width: 90 Class midpoints: 1004.5, 1094.5, 1184.5, 1274.5, 1364.5,1454.5 Class boundaries: 959.5, 1049.5, 1139.5, 1229.5, 1319.5, 1409.5, 1499.5 7. Class width: 100 Class midpoints: 49.5, 149.5, 249.5, 349.5, 449.5, 549.5, 649.5 Class boundaries: –0.5, 99.5, 199.5, 299.5, 399.5, 499.5, 599.5, 699.5 8. Class width: 100 Class midpoints: 149.5, 249.5, 349.5, 449.5, 549.5 Class boundaries: 99.5, 199.5, 299.5, 399.5, 499.5, 599.5 9. No. The maximum frequency is in the first class instead of being near the middle, so the frequencies below the maximum do not mirror those above the maximum. 10. No. The frequencies start high and then decrease, so the frequencies below the maximum do not mirror those above the maximum. 11. By symmetry, the last three frequencies would be 18, 12, and 2. The middle frequency would be 153  2 18  2 12   2  2   89. 12. No. The maximum frequency is in the second class instead of being near the middle, and the first two frequencies are much greater than the last two frequencies, so the frequencies below the maximum do not mirror those above the maximum. Copyright © 2018 Pearson Education, Inc. 8 Chapter 2: Exploring Data with Tables and Graphs 13. The pulse rates appear to have a distribution that is approximately normal. Pulse Rate (Male) 40–49 50–59 60–69 70–79 80–89 90–99 100–109 Frequency 2 23 53 43 25 5 2 14. The pulse rates appear to have a distribution that is approximately normal. Pulse Rate (Female) 30–39 40–49 50–59 60–69 70–79 80–89 90–99 100–109 Frequency 1 1 17 33 41 37 13 4 15. The verbal IQ scores appear to have a distribution that is approximately normal. IQ (Verbal) 50–59 60–69 70–79 80–89 90–99 100–109 110–119 120–129 Frequency 3 8 13 26 18 6 2 2 16. The verbal IQ scores appear to have a distribution that is approximately normal. IQ (Verbal) 60–69 70–79 80–89 90–99 100–109 Frequency 1 7 8 4 1 Copyright © 2018 Pearson Education, Inc. Section 2-1: Frequency Distributions for Organizing and Summarizing Data 9 17. Yes. The distribution appears to be approximately normal. Red Blood Cell Count (Males) 3.00–3.49 3.50–3.99 4.00–4.49 4.50–4.99 5.00–5.49 5.50–5.99 Frequency 1 16 29 57 44 6 18. Yes. The distribution appears to be approximately normal. Red Blood Cell Count (Females) 3.00–3.49 3.50–3.99 4.00–4.49 4.50–4.99 5.00–5.49 5.50–5.99 6.00–6.49 Frequency 1 20 85 33 7 0 1 Weight (kg) in September 50–59 60–69 70–79 80–89 90–99 Frequency 2 12 11 3 4 Weight (kg) in April 40–49 50–59 60–69 70–79 80–89 90–99 100–109 Frequency 2 20 28 8 7 1 1 19. 20. Copyright © 2018 Pearson Education, Inc. 10 Chapter 2: Exploring Data with Tables and Graphs 21. The frequency distribution suggests that the reported heights were rounded with disproportionately many 0s and 5s. This suggests that the results are not very accurate. Last Digit 0 1 2 3 4 5 6 7 8 9 Frequency 9 2 1 3 1 15 2 0 3 1 22. The frequency distribution suggests that the reported weights were not rounded since the last digits seem equally distributed. Last Digit 0 1 2 3 4 5 6 7 8 9 Frequency 4 3 6 4 4 5 7 7 6 4 23. The two distributions differ substantially. The presence of cotinine appears to be much higher for smokers than for nonsmokers exposed to smoke. Cotinine (ng/mL) 0–99 100–199 200–299 300–399 400–499 500–599 Smokers 27.5% 30.0% 35.0% 2.5% 5.0% 0% Copyright © 2018 Pearson Education, Inc. Nonsmokers Exposed to Smoke 85.0% 5.0% 2.5% 2.5% 0% 5.0% Section 2-1: Frequency Distributions for Organizing and Summarizing Data 11 24. The two distributions show moderate difference. It appears females have slightly higher pulse rates. Pulse Rate 0–99 100–199 200–299 300–399 400–499 500–599 600–699 Males 0.7% 33.3% 58.8% 6.5% 0% 0% 0.7% Females 0% 16.7% 61.3% 18.7% 0% 1.3% 2% 25. Cotinine (Nonsmokers Exposed to Smoke in ng/mL) Cumulative Frequency Less than 100 34 Less than 200 36 Less than 300 37 Less than 400 38 Less than 500 38 Less than 600 40 26. Brain Volume (cm3) Less than 1049 Less than 1139 Less than 1229 Less than 1319 Less than 1409 Less than 1499 Frequency 6 13 16 18 19 20 27. a. The values of 551 and 543 are clearly outliers; the values of 384, 241, 197, and 178 could also be outliers. b. The number of classes increases from six to ten. The outlier can greatly increase the number of classes. If there are too many classes, we might use a larger class width with the effect that the true nature of the distribution may be hidden. Cotinine (Nonsmokers Exposed to Smoke in ng/mL) 0–99 100–199 200–299 300–399 400–499 500–599 600–699 700–799 800–899 900–999 Frequency 34 2 1 1 0 2 0 0 0 1 Copyright © 2018 Pearson Education, Inc. 12 Chapter 2: Exploring Data with Tables and Graphs Section 2-2: Histograms 1. It is easier to see the distribution of the data by examining the graph of the histogram than by examining the numbers in a frequency distribution. 2. Not necessarily. Because the sample subjects themselves chose to be included, the voluntary response sample might not be representative of the population. 3. With a data set that is so small, the true nature of the distribution cannot be seen with a histogram. 4. No, the term “normal distribution” has a different meaning than the term “normal” that is used in ordinary speech. A normal distribution will have a histogram that is approximately bell shaped. Determining whether a histogram fits the bell shape would be subjective. 5. approximately 50 6. Class width: 0.5 mm, lower limit of first class: 2.0 mm, upper limit of first class: 2.4 mm 7. The largest possible value would be approximately 4.5 mm, which would not be an outlier. 8. The sample does not appear to be from a normal distribution, since it is not symmetric about the middle class. 9. The pulse rates of males appear to have a distribution that is approximately normal. Histogram for Exercise 9 Histogram for Exercise 10 60 40 50 30 Frequency Frequency 40 30 20 20 10 10 0 30 40 50 60 70 80 90 0 100 110 30 40 50 Pulse Rate (M ale) 60 70 80 90 100 110 Pulse Rate (Female) 10. The pulse rates of females appear to have a distribution that is approximately normal. 11. The IQ scores appear to have a distribution that is approximately normal. Histogram for Exercise 11 Histogram for Exercise 12 9 25 8 7 20 Frequency Frequency 6 15 10 5 4 3 2 5 1 0 60 80 100 120 0 60 Verbal IQ (Low Lead) 70 80 90 100 Verbal IQ (High Lead) 12. The IQ scores appear to have a distribution that is approximately normal. Copyright © 2018 Pearson Education, Inc. 110 Section 2-2: Histograms 13 13. Yes. The red blood cell counts appear to have a distribution that is very approximately normal, although some might describe the distribution as being left-skewed instead of normal. Histogram for Exercise 13 Histogram for Exercise 14 60 90 80 50 70 60 Frequency Frequency 40 30 50 40 20 30 20 10 0 10 3.0 3.5 4.0 4.5 5.0 5.5 0 6.0 3.0 3.5 Red Blood Cell Count (Female) 4.0 4.5 5.0 5.5 6.0 6.5 Red Blood Cell Count (M ale) 14. Yes. The red blood cell counts appear to have a distribution that is very approximately normal. 15. The histogram is shown below. Histogram for Exercise 15 Histogram for Exercise 16 14 20 12 15 Frequency Frequency 10 8 6 10 4 5 2 0 40 50 60 70 80 90 0 100 50 September Weight (kg) of M ales 60 70 80 90 100 April Weight (kg) of M ales 16. The histogram is shown above. 17. The histogram suggests that the reported heights were rounded with disproportionately many 0s and 5s. This suggests that the results are not very accurate. Histogram for Exercise 17 Histogram for Exercise 18 16 7 14 6 5 10 Frequency Frequency 12 8 6 3 2 4 1 2 0 4 0 1 2 3 4 5 6 7 8 9 0 Last Digit 0 1 2 3 4 5 Last Digit Copyright © 2018 Pearson Education, Inc. 6 7 8 9 14 Chapter 2: Exploring Data with Tables and Graphs 18. The histogram suggests that the reported weights were not rounded since the last digits seem equally distributed. 19. Only part (c) appears to represent data from a normal distribution. Part (a) has a systematic pattern that is not that of a straight line, part (b) has points that are not close to a straight-line pattern, and part (d) is really bad because it shows a systematic pattern and points that are not close to a straight-line pattern. Section 2-3: Graphs That Enlighten and Graphs That Deceive 1. The data set is too small for a graph to reveal important characteristics of the data. With such a small data set, it would be better to simply list the data or place them in a table. 2. No. If the sample is a bad sample, such as one obtained from voluntary responses, there are no graphs or other techniques that can be used to salvage the data. 3. No. Graphs should be constructed in a way that is fair and objective. The readers should be allowed to make their own judgments, instead of being manipulated by misleading graphs. 4. Center, variation, distribution, outliers, change in the characteristics of data over time. The time-series graph does the best job of giving us insight into the change in the characteristics of data over time. 5. The pulse rate of 36 beats per minute appears to be an outlier. 6. There do not appear to be any outliers. 7. The data are arranged in order from lowest to highest, as 36, 56, 56, and so on. 8. The two values closest to the middle are 72 mm Hg and 74 mm Hg. Copyright © 2018 Pearson Education, Inc. Section 2-3: Graphs That Enlighten and Graphs That Deceive 15 9. There was a steep jump in the first four years, but the numbers of triplets have shown a downward trend in the past several years. 8000 Number of Triplets 7000 6000 5000 4000 3000 2000 1000 0 1995.0 1997.5 2000.0 2002.5 2005.0 2007.5 2010.0 2012.5 Year 10. The number of fatalities is decreasing, most likely due to greater enforcement of DUI laws and greater public awareness campaigns. 6.5 6.0 Fatalities 5.5 5.0 4.5 4.0 3.5 3.0 1990 1995 2000 2005 2010 2015 Year 11. Misconduct includes fraud, duplication, and plagiarism, so it does appear to be a major factor. 900 Number of Retractions 800 700 600 500 400 300 200 100 0 Fraud Error Duplication Other Copyright © 2018 Pearson Education, Inc. Plagiarism 16 Chapter 2: Exploring Data with Tables and Graphs 12. The overwhelming response was that thank-you notes should be sent to everyone who is met during a job interview. Given what is at stake, that seems like a wise strategy. 400 Number 300 200 100 0 Everyone M ost Senior M ost Time Best Conversation Don’t Send 13. 14. 15. The distribution appears to be roughly bell-shaped, so the distribution is approximately normal. 60 50 Frequency 40 30 20 10 0 34.5 44.5 54.5 64.5 74.5 84.5 Pulse Rate (M ale) Copyright © 2018 Pearson Education, Inc. 94.5 104.5 Section 2-4: Scatterplots, Correlation, and Regression 17 16. The distribution appears to be roughly bell-shaped, so the distribution is approximately normal. 40 Frequency 30 20 10 0 24.5 34.5 44.5 54.5 64.5 74.5 84.5 94.5 104.5 114.5 Pulse Rate (Female) 17. Because the vertical scale starts with a frequency of 200 instead of 0, the difference between the “no” and “yes” responses is greatly exaggerated. The graph makes it appear that about five times as many respondents said “no,” when the ratio is actually a little less than 2.5 to 1. 18. The two costs are one-dimensional in nature, but the baby bottles are three-dimensional objects. The $4500 cost isn’t even twice the $2600 cost, but the baby bottles make it appear that the larger cost is about five times the smaller cost. Section 2-4: Scatterplots, Correlation, and Regression 1. The term linear refers to a straight line, and r measures how well a scatterplot of the sample paired data fits a straight-line pattern. 2. No. Finding the presence of a statistical correlation between two variables does not justify any conclusion that one of the variables is a cause of the other. 3. A scatterplot is a graph of paired  x, y  quantitative data. It helps us by providing a visual image of the data plotted as points, and such an image is helpful in enabling us to see patterns in the data and to recognize that there may be a correlation between the two variables. 4. a. 1 b. 0 c. 0 d. –1 5. There does not appear to be a linear correlation between brain volume and IQ score. Scatterplot for Exercise 5 Scatterplot for Exercise 6 115 450 400 110 350 IQ Weight (lb) 105 100 300 250 200 150 95 100 90 1000 1100 1200 1300 1400 Volume 6. 25 30 35 40 Chest (in.) There does appear to be a linear correlation between chest sizes and weights of bears. Copyright © 2018 Pearson Education, Inc. 45 50 55 18 Chapter 2: Exploring Data with Tables and Graphs 7. There does not appear to be a linear correlation between body temperature at 8 AM on one day and at 8 AM on the following day. Scatterplot for Exercise 7 Scatterplot for Exercise 8 77 97.75 76 Height of first son (in.) Day 2 97.50 97.25 97.00 96.75 96.50 96.5 75 74 73 72 71 97.0 97.5 98.0 70 98.5 73 74 Day 1 8. 75 76 77 78 79 Height of father (in.) There does not appear to be a linear correlation between heights of fathers and the heights of their first sons. 9. With n  5 pairs of data, the critical values are 0.878. Because r  0.127 is between –0.878 and 0.878, evidence is not sufficient to conclude that there is a linear correlation. 10. With n  7 pairs of data, the critical values are 0.754. Because r  0.980 is in the right tail region beyond 0.754, there are sufficient data to conclude that there is a linear correlation. 11. With n  7 pairs of data, the critical values are 0.754. Because r  0.502 is between –0.754 and 0.754, evidence is not sufficient to conclude that there is a linear correlation. 12. With n  10 pairs of data, the critical values are 0.632. Because r  0.017 is between –0.632 and 0.632, evidence is not sufficient to conclude that there is a linear correlation. Chapter Quick Quiz 1. The class width is 0.12  0.08  0.04. 2. The class boundaries are 0.075 and 0.115. 3. No, it is impossible to determine the original values. 4. 16, 17, 18, 18, 19 7. time-series graph 5. bell-shaped 8. scatterplot 6. variation 9. Pareto chart 10. A frequency distribution is in the format of a table, but a histogram is a graph. Review Exercises 1. Temperature (F) 97.0–97.4 97.5–97.9 98.0–98.4 98.5–98.9 99.0–99.4 Frequency 2 4 7 5 2 Copyright © 2018 Pearson Education, Inc. Review Exercises 19 2. Yes, the data appear to be from a population with a normal distribution because the bars start low and reach a maximum, then decrease, and the left half of the histogram is approximately a mirror image of the right half. 7 6 Frequency 5 4 3 2 1 0 97.2 97.7 98.2 98.7 99.2 Body Temperature (Fahrenheit) 3. By using fewer classes, the histogram does a better job of illustrating the distribution. 4. There are no outliers. 5. Yes. There is a pattern suggesting that there is a relationship. 32.5 30.0 Neck Size (in.) 27.5 25.0 22.5 20.0 17.5 15.0 100 150 200 250 300 350 400 450 Weight (lb) 6. a. time-series graph b. scatterplot c. Pareto chart 7. By using a vertical scale that starts at 45% instead of 0%, the difference is greatly exaggerated. The graph creates the false impression that male enrollees outnumber female enrollees by a ratio of about 3:1, but the actual percentages of 53% and 47% are much closer than that. Copyright © 2018 Pearson Education, Inc. 20 Chapter 2: Exploring Data with Tables and Graphs Cumulative Review Exercises 1. Grooming Time (min) 0–9 10–19 20–29 30–39 40–49 Frequency 2 3 9 4 2 2. The histogram is approximately bell-shaped. The frequencies increase to a maximum and then decrease, and the left half of the histogram is roughly a mirror image of the right half. The data do appear to be from a population with a normal distribution. 9 8 Frequency 7 6 5 4 3 2 1 0 0 10 20 30 40 50 Grooming Time (min) 3. 4. There are disproportionately many last digits of 0 and 5. Fourteen of the 20 times have last digits of 0 or 5. It appears that the subjects reported their results and they tended to round the results. The data do not appear to be very accurate. Last Digit 0 1 2 3 4 5 6 7 8 9 Frequency 5 0 2 0 1 9 0 2 1 0 Copyright © 2018 Pearson Education, Inc. Cumulative Review Exercises 21 5. a. ratio b. continuous c. No. The grooming times are quantitative data. d. statistic 6. The scatterplot helps address the issue of whether there is a correlation between heights of mothers and heights of their daughters. The scatterplot does not reveal a clear pattern suggesting that there is a correlation. 71 Daughter’s Height (in.) 70 69 68 67 66 65 58 60 62 64 66 68 M other’s Height (in.) Copyright © 2018 Pearson Education, Inc.