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

Solution Manual for Biostatistics for the Biological and Health Sciences, 2nd Edition breaks down difficult textbook problems into simple solutions, making your study time more effective.

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    S OLUTIONS M ANUAL J AMES L APP Colorado Mesa University B IOSTATISTICS FOR THE B IOLOGICAL AND H EALTH S CIENCES SECOND EDITION Marc Triola New York University School of Medicine Mario F. Triola Dutchess Community College Jason Roy University of Pennsylvania Perelman School of Medicine

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    CONTENTS Chapter 1: Introduction To Statistics Section 1-1: Statistical and Critical Thinking............................................................................1 Section 1-2: Types of Data ........................................................................................................2 Section 1-3: Collecting Sample Data.........................................................................................3 Chapter Quick Quiz ...................................................................................................................5 Review Exercises .......................................................................................................................5 Cumulative Review Exercises ...................................................................................................6 Chapter 2: Exploring Data with Tables and Graphs Section 2-1: Frequency Distributions for Organizing and Summarizing Data..........................7 Section 2-2: Histograms...........................................................................................................12 Section 2-3: Graphs That Enlighten and Graphs That Deceive...............................................14 Section 2-4: Scatterplots, Correlation, and Regression ...........................................................17 Chapter Quick Quiz .................................................................................................................18 Review Exercises .....................................................................................................................18 Cumulative Review Exercises .................................................................................................20 Chapter 3: Describing, Exploring, and Comparing Data Section 3-1: Measures of Center..............................................................................................23 Section 3-2: Measures of Variation .........................................................................................27 Section 3-3: Measures of Relative Standing and Boxplots......................................................31 Chapter Quick Quiz .................................................................................................................34 Review Exercises .....................................................................................................................35 Cumulative Review Exercises .................................................................................................35 Chapter 4: Probability Section 4-1: Basic Concepts of Probability .............................................................................37 Section 4-2: Addition Rule and Multiplication Rule ...............................................................38 Section 4-3: Complements, Conditional Probability, and Bayes’ Theorem............................41 Section 4-4: Risks and Odds ....................................................................................................42 Section 4-5: Rates of Mortality, Fertility, and Morbidity........................................................44 Section 4-6: Counting ..............................................................................................................46 Chapter Quick Quiz .................................................................................................................48 Review Exercises .....................................................................................................................48 Cumulative Review Exercises .................................................................................................49 Chapter 5: Discrete Probability Distributions Section 5-1: Probability Distributions .....................................................................................51 Section 5-2: Binomial Probability Distributions......................................................................52 Section 5-3: Poisson Probability Distributions ........................................................................55 Chapter Quick Quiz .................................................................................................................57 Review Exercises .....................................................................................................................58 Cumulative Review Exercises .................................................................................................59

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    Chapter 6: Normal Probability Distributions Section 6-1: The Standard Normal Distribution ......................................................................61 Section 6-2: Real Applications of Normal Distributions.........................................................62 Section 6-3: Sampling Distributions and Estimators ...............................................................65 Section 6-4: The Central Limit Theorem.................................................................................69 Section 6-5: Assessing Normality............................................................................................72 Section 6-6: Normal as Approximation to Binomial ...............................................................75 Chapter Quick Quiz .................................................................................................................77 Review Exercises .....................................................................................................................77 Cumulative Review Exercises .................................................................................................79 Chapter 7: Estimating Parameters and Determining Sample Sizes Section 7-1: Estimating a Population Proportion.....................................................................81 Section 7-2: Estimating a Population Mean.............................................................................85 Section 7-3: Estimating a Population Standard Deviation or Variance...................................89 Section 7-4: Bootstrapping: Using Technology for Estimates ................................................92 Chapter Quick Quiz .................................................................................................................94 Review Exercises .....................................................................................................................94 Cumulative Review Exercises .................................................................................................96 Chapter 8: Hypothesis Testing Section 8-1: Basics of Hypothesis Testing ..............................................................................97 Section 8-2: Testing a Claim About a Proportion....................................................................99 Section 8-3: Testing a Claim About a Mean..........................................................................104 Section 8-4: Testing a Claim About a Standard Deviation or Variance ................................108 Chapter Quick Quiz ...............................................................................................................110 Review Exercises ...................................................................................................................111 Cumulative Review Exercises ...............................................................................................112 Chapter 9: Chapter 9: Inferences from Two Samples Section 9-1: Two Proportions ................................................................................................115 Section 9-2: Two Means: Independent Samples....................................................................124 Section 9-3: Two Dependent Samples (Matched Pairs) ........................................................130 Section 9-4: Two Variances or Standard Deviations.............................................................134 Chapter Quick Quiz ...............................................................................................................136 Review Exercises ...................................................................................................................137 Cumulative Review Exercises ...............................................................................................138 Chapter 10: Correlation and Regression Section 10-1: Correlation.......................................................................................................141 Section 10-2: Regression .......................................................................................................148 Section 10-3: Prediction Intervals and Variation...................................................................156 Section 10-4: Multiple Regression.........................................................................................159 Section 10-5: Dummy Variables and Logistic Regression ....................................................161 Chapter Quick Quiz ...............................................................................................................162 Review Exercises ...................................................................................................................162 Cumulative Review Exercises ...............................................................................................164

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    Chapter 11: Goodness-of-Fit and Contingency Tables Section 11-1: Goodness-of-Fit...............................................................................................167 Section 11-2: Contingency Tables .........................................................................................170 Chapter Quick Quiz ...............................................................................................................174 Review Exercises ...................................................................................................................175 Cumulative Review Exercises ...............................................................................................176 Chapter 12: Analysis of Variance Section 12-1: One-Way ANOVA ..........................................................................................177 Section 12-2: Two-Way ANOVA .........................................................................................178 Chapter Quick Quiz ...............................................................................................................179 Review Exercises ...................................................................................................................180 Cumulative Review Exercises ...............................................................................................180 Chapter 13: Nonparametric Tests Section 13-2: Sign Test..........................................................................................................183 Section 13-3: Wilcoxon Signed-Ranks Test for Matched Pairs ............................................184 Section 13-4: Wilcoxon Rank-Sum Test for Two Independent samples...............................185 Section 13-5: Kruskal-Wallis Test for Three or More Samples ............................................188 Section 13-6: Rank Correlation .............................................................................................190 Chapter Quick Quiz ...............................................................................................................191 Review Exercises ...................................................................................................................192 Cumulative Review Exercises ...............................................................................................192 Chapter 14: Survival Analysis Section 14-1: Life Tables.......................................................................................................195 Section 14-2: Kaplan-Meier Survival Analysis .....................................................................197 Chapter Quick Quiz ...............................................................................................................198 Review Exercises ...................................................................................................................199 Cumulative Review Exercises ...............................................................................................200

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    Section 1-1: Statistical and Critical Thinking 1 Chapter 1: Introduction to Statistics Section 1-1: Statistical and Critical Thinking 1. The respondents are a voluntary response sample or a self-selected sample. Because those with strong interests in the topic are more likely to respond, it is very possible that their responses do not reflect the opinions or behavior of the general population. 2. a. The sample consists of the 1046 adults who were surveyed. The population consists of all adults. b. When asked, respondents might be inclined to avoid the shame of the unhealthy habit of not washing their hands, so the reported rate of 70% might well be much higher than it is in reality. It is generally better to observe or measure human behavior than to ask subjects about it. 3. Statistical significance is indicated when methods of statistics are used to reach a conclusion that a treatment is effective, but common sense might suggest that the treatment does not make enough of a difference to justify its use or to be practical. Yes, it is possible for a study to have statistical significance, but not practical significance. 4. No. Correlation does not imply causation. The example illustrates a correlation that is clearly not the result of any interaction or cause effect relationship between deaths in swimming pools and power generated from nuclear power plants. 5. Yes, there does appear to be a potential to create a bias. 6. No, there does not appear to be a potential to create a bias. 7. No, there does not appear to be a potential to create a bias. 8. Yes, there does appear to be a potential to create a bias. 9. The sample is a voluntary response sample and has strong potential to be flawed. 10. The samples are voluntary response samples and have potential for being flawed, but this approach might be necessary due to ethical considerations involved in randomly selecting subjects and somehow imposing treatments on them. 11. The sampling method appears to be sound. 12. The sampling method appears to be sound. 13. With only a 1% chance of getting such results with a program that has no effect, the program appears to have statistical significance. Also, because the average loss of 22 pounds does seem substantial, the program appears to also have practical significance. 14. Because there is a 0.3% chance of getting such results by chance, the increase in scores does appear to have statistical significance. The typical increase of 5 points suggests that the course does have practical significance. The course does appear to be successful. 15. Because there is a 19% chance of getting that many girls by chance, the method appears to lack statistical significance. The result of 1020 girls in 2000 births (51% girls) is above the approximately 50% rate expected by chance, but it does not appear to be high enough to have practical significance. Not many couples would bother with a procedure that raises the likelihood of a girl from 50% to 51%. 16. Because there is a 15% chance of getting such results with a medication that has no effect, the medication does not appear to have statistical significance. Because the average decrease is only 2 mm Hg, the medication does not appear to have practical significance. 17. Yes. Each column of 8 AM and 12 AM temperatures is recorded from the same subject, so each pair is matched. 18. No. The source is from university researchers who do not appear to gain from distorting the data. 19. The data can be used to address the issue of whether there is a correlation between body temperatures at 8 AM and at 12 AM. Also, the data can be used to determine whether there are differences between body temperatures at 8 AM and at 12 AM.

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    2 Chapter 1: Introduction to Statistics 20. Because the differences could easily occur by chance (with a 64% chance), the differences do not appear to have statistical significance. 21. No. The white blood cell counts measure a different quantity than the red blood cell counts, so their differences are meaningless. 22. The issue that can be addressed is whether there is a correlation, or association, between white blood cell counts and red blood cell counts. 23. No. The National Center for Health Statistics has no reason to collect or present the data in a way that is biased. 24. No. Correlation does not imply causation, so a statistical correlation between white blood cell counts and red blood cell counts should not be used to conclude that higher white blood cell counts are the cause of higher red blood cell counts. 25. It is questionable that the sponsor is the Idaho Potato Commission and the favorite vegetable is potatoes. 26. The sample is a voluntary response sample, so there is a good chance that the results do not reflect the larger population of people who have a water preference. 27. The correlation, or association, between two variables does not mean that one of the variables is the cause of the other. Correlation does not imply causation. Clearly, cheese consumption is not directly related in any way to bedsheet deaths. 28. The sponsor of the poll is an electronic cigarette maker, so the sponsor does have an interest in the poll results. The source is questionable. 29. a.   0.45 3014 1356.3  adults b. No. Because the result is a count of the people among the 3014 who were surveyed, the result must be a whole number. c. 1356 adults d. 1206 0.40, 1206 1808   or 40% 30. a.   0.19 1543 293.17  women b. No. Because the result is a count of women among the 1543 who were surveyed, the result must be a whole number. c. 293 women d. 219 0.15, 1458  or 15% e. Interpretations of a “typical” week and what it means to “kick back and relax” might vary considerably by different survey respondents. The survey might be improved by asking about behavior within “the past seven days” instead of a “typical” week. Instead of “kick back and relax,” respondents might be surveyed about specific behavior, such as reading, taking a nap, watching television, listening to music, or going for a walk. 31. The wording of the question is biased and tends to encourage negative responses. The sample size of 20 is too small. Survey respondents are self-selected instead of being randomly selected by the newspaper. If 20 readers respond, the percentages should be multiples of 5, so 87% and 13% are not possible results. 32. All percentages of success should be multiples of 5. The given percentages cannot be correct. Section 1-2: Types of Data 1. a. The sample is the 1020 adults who were surveyed. The population is all adults in the United States. b. statistic 2. a. ratio b. discrete c. categorical

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    Section 1-3: Collecting Sample Data 3 3. a. quantitative b. categorical c. categorical d. quantitative 4. Only part (a) describes discrete data. 5. statistic 6. statistic 7. statistic 8. parameter 9. statistic 10. statistic 11. parameter 12. parameter 13. continuous 14. discrete 15. continuous 16. continuous 17. discrete 18. continuous 19. discrete 20. discrete 21. ratio 22. ordinal 23. interval 24. nominal 25. nominal 26. ratio 27. ordinal 28. ordinal 29. The numbers are not counts or measures of anything. They are at the nominal level of measurement, and it makes no sense to compute the average (mean) of them. 30. The digits are not counts or measures of anything. They are at the nominal level of measurement and it makes no sense to calculate their average (mean). 31. The temperatures are at the interval level of measurement. Because there is no natural starting point with 0 F  representing “no heat,” ratios such as “twice” make no sense, so it is wrong to say that it is twice as warm at the author’s home as it is in Auckland, New Zealand. 32. The ranks are at the ordinal level of measurement. Differences between the universities cannot be determined, so there is no way to know whether the difference between Princeton and Harvard is the same as the difference between Yale and Columbia. 33. a. Continuous, because the number of possible values is infinite and not countable. b. Discrete, because the number of possible values is finite. c. Discrete, because the number of possible values is finite. d. Discrete, because the number of possible values is infinite and countable. Section 1-3: Collecting Sample Data 1. The study is an experiment because subjects were given treatments. 2. The subjects in the study did not know whether they were taking a placebo or the paracetamol medication, and those who administered the pills also did not know. 3. The group sample sizes of 547, 550, and 546 are all large so that the researchers could see the effects of the paracetamol treatment. 4. The sample appears to be a convenience sample. Given that the subjects were randomly assigned to the three different treatment groups, it appears that the results of the study are good because they are not likely to be distorted from bias, but we should investigate the sample groups to ensure that they are not fundamentally different from the population. 5. The sample appears to be a convenience sample. By e-mailing the survey to a readily available group of Internet users, it was easy to obtain results. Although there is a real potential for getting a sample group that is not representative of the population, indications of which ear is used for cell phone calls and which hand is dominant do not appear to be factors that would be distorted much by a sample bias.

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    4 Chapter 1: Introduction to Statistics 6. The study is an observational study because the subjects were not given any treatment. 7. With 717 responses, the response rate is 14%, which does appear to be quite low. In general, a very low response rate creates a serious potential for getting a biased sample that consists of those with a special interest in the topic. 8. Answers vary, but the following are good possibilities. a. Obtain a printed copy of the class roster, assign consecutive numbers (integers), then use a computer to randomly generate six of those numbers. b. Select every third student leaving class until six students are chosen. c. Randomly select three males and three females. d. Randomly select a row, and then select the students in that row. (Use only the first six to meet the requirement of a sample of size six.) 9. systematic 10. convenience 11. random 12. stratified 13. cluster 14. random 15. stratified 16. systematic 17. random 18. cluster 19. convenience 20. systematic 21. Observational study. The sample is a convenience sample consisting of subjects who decided themselves to respond. Such voluntary response samples have a high chance of not being representative of the larger population, so the sample may well be biased. The question was posted in an electronic edition of a newspaper, so the sample is biased from the beginning. 22. Experiment. The sample subjects consist of male physicians only. It would have been better to include females. Also, it would be better to include male and females who are not physicians. 23. Experiment. This experiment would create an extremely dangerous and illegal situation that has a real potential to result in injury or death. It’s difficult enough to drive in New York City while being completely sober. 24. Observational study. The sample of four males and four females is too small. 25. Experiment. The biased sample created by using subjects from New York City cannot be fixed by using a larger sample. The larger sample will still be a biased sample that is not representative of subjects in the United States. 26. Experiment. Calling the subjects and asking them to report their weights has a high risk of getting results that do not reflect the actual weights. It would have been much better to somehow measure the weights instead of asking the subjects to report them. 27. Observational study. Respondents who have been convicted of felonies are not likely to respond honestly to the second question. The survey will suffer from a “social desirability bias” because subjects will tend to respond in ways that will be viewed favorably by those conducting the survey. 28. Observational study. The number of responses is very small, and the response rate of only 1.52% is far too small. With such a low response rate, there is a real possibility that the sample of respondents is biased and consists only of those with special interests in the survey topic. 29. prospective study 30. retrospective study 31. cross-sectional study 32. prospective study 33. matched pairs design 34. randomized block design 35. completely randomized design 36. matched pairs design

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    Chapter Quick Quiz 5 37. Prospective: The experiment was begun and results were followed forward in time. Randomized: Subjects were assigned to the different groups through a process of random selection, whereby they had the same chance of belonging to each group. Double-blind: The subjects did not know which of the three groups they were in, and the people who evaluated results did not know either. Placebo-controlled: There was a group of subjects who were given a placebo; by comparing the placebo group to the two treatment groups, the effects of the treatments might be better understood. 38. a. Not a simple random sample, but it is a random sample. b. Simple random sample and also a random sample. c. Not a simple random sample and not a random sample. Chapter Quick Quiz 1. No. The numbers do not measure or count anything. 2. nominal 3. continuous 4. quantitative data 5. ratio 6. no 7. no 8. statistic 9. observational study 10. false Review Exercises 1. a. discrete b. ratio c. stratified d. cluster e. The mailed responses would be a voluntary response sample, so those with strong opinions are more likely to respond. It is very possible that the results do not reflect the true opinions of the population of all customers. 2. The survey was sponsored by the American Laser Centers, and 24% said that the favorite body part is the face, which happens to be a body part often chosen for some type of laser treatment. The source is therefore questionable. 3. The sample is a voluntary response sample, so the results are questionable. 4. a. It uses a voluntary response sample, and those with special interests are more likely to respond, so it is very possible that the sample is not representative of the population. b. Because the statement refers to 72% of all Americans, it is a parameter (but it is probably based on a 72% rate from the sample, and the sample percentage is a statistic). c. observational study 5. a. If they have no fat at all, they have 100% less than any other amount with fat, so the 125% figure cannot be correct. b.   0.58 1182 685.56,  so 686 c. 331 1182 0.28,  or 28% 6. Only part (c) is a simple random sample. 7. Because there is only a 4% chance of getting the results by chance, the method appears to have statistical significance. The result of 112 girls in 200 births is above the approximately 50% rate expected by chance, but it does not appear to be high enough to have practical significance. Not many couples would bother with a procedure that raises the likelihood of a girl from 50% to 56%. 8. a. random b. stratified c. nominal d. Statistic, because it is based on a sample. e. The mailed responses would be a voluntary response sample. Those with strong opinions about the topic would be more likely to respond, so it is very possible that the results would not reflect the true opinions of the population of all adults.

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    6 Chapter 1: Introduction to Statistics 9. a. systematic; It is likely to result in a representative sample. b. random; It is likely to result in a representative sample. c. cluster; It is likely to result in a representative sample. d. stratified; It is likely to result in a representative sample. e. convenience; It is very possible that the sample is not representative. 10. a.   0.52 1500 780  adults b. 345 0.23 1500  or 23% c. Men: 727 0.485 1500  or 48.5%; Women: 773 0.515 1500  or 51.5% d. No, although this is a subjective judgment. e. No, although this is a subjective judgment. Cumulative Review Exercises 1. The mean is 3600 1700 4000 3900 3100 3800 2200 3000 3162.5 8         grams. The weights all end with 00, suggesting that all of the weights are rounded to the hundreds place, so that the last two digits are always 00. 2. 6 0.5 0.015625  3. 272 176 16, 6   which is an unusually high value. 4. 98.2 98.6 6.64 0.62 106    5. 2 2 1.96 0.25 1067 0.03   6. 4000 1700 575 4   grams 7.   2 3600 3162.5 27, 343.75 7   grams 2 8.       2 2 2 98.4 98.6 98.6 98.6 98.8 98.6 0.04 0.20 3 1         9. 8 0.4 0.00065536  10. 11 9 31, 381,059, 609 (or about 31, 381,060, 000)  11. 14 6 78,364,164,096 (or about 78,364,164,000)  12. 12 0.3 0.000000531441 

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    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) Relative Frequency 0–99 27.5% 100–199 30.0% 200–299 35.0% 300–399 2.5% 400–499 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.

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    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) Frequency 40–49 2 50–59 23 60–69 53 70–79 43 80–89 25 90–99 5 100–109 2 14. The pulse rates appear to have a distribution that is approximately normal. Pulse Rate (Female) Frequency 30–39 1 40–49 1 50–59 17 60–69 33 70–79 41 80–89 37 90–99 13 100–109 4 15. The verbal IQ scores appear to have a distribution that is approximately normal. IQ (Verbal) Frequency 50–59 3 60–69 8 70–79 13 80–89 26 90–99 18 100–109 6 110–119 2 120–129 2 16. The verbal IQ scores appear to have a distribution that is approximately normal. IQ (Verbal) Frequency 60–69 1 70–79 7 80–89 8 90–99 4 100–109 1

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    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) Frequency 3.00–3.49 1 3.50–3.99 16 4.00–4.49 29 4.50–4.99 57 5.00–5.49 44 5.50–5.99 6 18. Yes. The distribution appears to be approximately normal. Red Blood Cell Count (Females) Frequency 3.00–3.49 1 3.50–3.99 20 4.00–4.49 85 4.50–4.99 33 5.00–5.49 7 5.50–5.99 0 6.00–6.49 1 19. Weight (kg) in September Frequency 50–59 2 60–69 12 70–79 11 80–89 3 90–99 4 20. Weight (kg) in April Frequency 40–49 2 50–59 20 60–69 28 70–79 8 80–89 7 90–99 1 100–109 1

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    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 Frequency 0 9 1 2 2 1 3 3 4 1 5 15 6 2 7 0 8 3 9 1 22. The frequency distribution suggests that the reported weights were not rounded since the last digits seem equally distributed. Last Digit Frequency 0 4 1 3 2 6 3 4 4 4 5 5 6 7 7 7 8 6 9 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) Smokers Nonsmokers Exposed to Smoke 0–99 27.5% 85.0% 100–199 30.0% 5.0% 200–299 35.0% 2.5% 300–399 2.5% 2.5% 400–499 5.0% 0% 500–599 0% 5.0%

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    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 Males Females 0–99 0.7% 0% 100–199 33.3% 16.7% 200–299 58.8% 61.3% 300–399 6.5% 18.7% 400–499 0% 0% 500–599 0% 1.3% 600–699 0.7% 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 (cm 3 ) Frequency Less than 1049 6 Less than 1139 13 Less than 1229 16 Less than 1319 18 Less than 1409 19 Less than 1499 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) Frequency 0–99 34 100–199 2 200–299 1 300–399 1 400–499 0 500–599 2 600–699 0 700–799 0 800–899 0 900–999 1

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