12Chapter 1What Is Statistics?Is the amount of time spent online using the Internet approximately normally distributed?Why?1.11The following results on summations will help us in calculating the sample variance s2.Forany constant c,Si=1Use (a), (b), and (c) to show that-52n-141.12Use theresultofExercise1.11 to calculate sfor the n =6 sample measurements1,4,2,1,3and 3.1.13Refer to Exercise 1.2.Calculate y and s for the data given.abCalculate the interval y±ks fork =1, 2, and 3.Count the number of measurements thatfall within each interval and compare this result with the number that you would expectaccordingtotheempirical rule.1.14Refer to Exercise 1.3 and repeat parts (a) and (b) of Exercise 1.13.1.15Refer to Exercise 1.4 and repeat parts (a) and (b) of Exercise 1.13.1.16In Exercise 1.4, there is one extremely large value (11.88). Eliminate this value and calculatey and sfor the remaining 39 observations.Also, calculatetheintervalsy ±ks fork=12, and 3; count the number of measurements in each; then compare these results with thosepredicted by the empirical rule.Comparethe answersheretothosefound inExercise1.15Notetheeffectofasinglelargeobservationonyands.1.17The range ofa set of measurements is the difference between the largest and the smallest values.The empirical rule suggests that the standard deviation of a set of measurements may be roughlyapproximated by one-fourth of the range (that is,range/4).Calculate this approximation to sfor the data sets in Exercises1.2.1.3, and 1.4.Compare the result in each case to theactual,calculatedvalue ofs1.18The College Board's verbal and mathematics Scholastic Aptitude Tests are scored on a scale of200to800.Itseemsreasonabletoassumethatthedistributionoftestscoresareapproximatelynormallydistributedforbothtests.UsetheresultfromExercise1.17toapproximatethestandarddeviation for scores on the verbal test.1.19According to the Environmental Protection Agency, chloroform, which in its gaseous formis suspected to be a cancer-causing agent, is present in small quantities in all the country's240,000 public water sources.Ifthemean and standard deviationoftheamountsofchloroformpresent in water sources are 34 and 53 micrograms per liter (μg/L),respectively,explain whychloroformamountsdonothaveanormaldistribution.lRightCopyright 2011 Cengage LEhison
12 Chapter 1 What Is Statistics? c Is the amount of time spent online using the Internet approximately normally distributed? Why? 1.11 The following results on summations will help us in calculating the sample variance s 2 . For any constant c, a bn i=1 c = nc. b bn i=1 cyi = c bn i=1 yi . c bn i=1 (xi + yi) = bn i=1 xi + bn i=1 yi . Use (a), (b), and (c) to show that s 2 = 1 n − 1 bn i=1 (yi − y) 2 = 1 n − 1 ⎡ ⎣ bn i=1 y 2 i − 1 n Ibn i=1 yi P2 ⎤ ⎦. 1.12 Use the result of Exercise 1.11 to calculate s for the n = 6 sample measurements 1, 4, 2, 1, 3, and 3. 1.13 Refer to Exercise 1.2. a Calculate y and s for the data given. b Calculate the interval y ± ks for k = 1, 2, and 3. Count the number of measurements that fall within each interval and compare this result with the number that you would expect according to the empirical rule. 1.14 Refer to Exercise 1.3 and repeat parts (a) and (b) of Exercise 1.13. 1.15 Refer to Exercise 1.4 and repeat parts (a) and (b) of Exercise 1.13. 1.16 In Exercise 1.4, there is one extremely large value (11.88). Eliminate this value and calculate y and s for the remaining 39 observations. Also, calculate the intervals y ± ks for k = 1, 2, and 3; count the number of measurements in each; then compare these results with those predicted by the empirical rule. Compare the answers here to those found in Exercise 1.15. Note the effect of a single large observation on y and s. 1.17 The range of a set of measurements is the difference between the largest and the smallest values. The empirical rule suggests that the standard deviation of a set of measurements may be roughly approximated by one-fourth of the range (that is, range/4). Calculate this approximation to s for the data sets in Exercises 1.2, 1.3, and 1.4. Compare the result in each case to the actual, calculated value of s. 1.18 The College Board’s verbal and mathematics Scholastic Aptitude Tests are scored on a scale of 200 to 800. It seems reasonable to assume that the distribution of test scores are approximately normally distributed for both tests. Use the result from Exercise 1.17 to approximate the standard deviation for scores on the verbal test. 1.19 According to the Environmental Protection Agency, chloroform, which in its gaseous form is suspected to be a cancer-causing agent, is present in small quantities in all the country’s 240,000 public water sources. If the mean and standard deviation of the amounts of chloroform present in water sources are 34 and 53 micrograms per liter (μg/L), respectively, explain why chloroform amounts do not have a normal distribution. Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it
1.4HowInferencesAreMade131.20Weekly maintenance costs for a factory,recorded over a long period of time and adjustedfor inflation, tend to have an approximately normal distribution with an average of S420 and astandarddeviation of$30.If $450 isbudgetedfor next week,what is an approximate probabilitythat this budgeted figure will be exceeded?1.21The manufacturer of a newfood additivefor beef cattle claims that 80% of the animals fed adiet including this additive should have monthly weight gains in excess of 20 pounds. A largesample of measurements on weight gains for cattle fed this diet exhibits an approximatelynormal distributionwith mean 22pounds and standarddeviation 2pounds.Do you think thesampleinformationcontradictsthemanufacturer'sclaim?(Calculatetheprobabilityofaweightgain exceeding 20 pounds.)1.4HowlnferencesAreMadeThe mechanism instrumental in making inferences canbe wellillustratedby analyzingourown intuitiveinference-makingprocedures.Supposethattwo candidates arerunning fora public office inour communityand that we wish to determine whether our candidate, Jones, is favored to win.Thepopulationof interestisthesetofresponsesfromall eligiblevoterswhowill voteonelection day,and we wish to determine whether thefraction favoring Jones exceeds.5.For the sake of simplicity,suppose that all eligible voters will go to the polls and thatwe randomly selecta sample of 20 from the courthouserosterof voters.All20 arecontacted and all favor Jones.What do you conclude about Jones's prospects forwinningtheelection?There is little doubt that most of us would immediately infer that Jones will win.This is an easy inference to make, but this inference itself is not our immediate goal.Rather, we wish to examine the mental processes that were employed in reaching thisconclusion about theprospective behavior of a largevotingpopulation based onasampleofonly20people.Winning means acquiring more than 50% of the votes.Did we conclude that Joneswould winbecause wethoughtthatthefractionfavoringJonesin the sample wasidentical to the fraction favoring Jones in the population? Weknow that this is prob-ably not true. A simple experiment will verify that the fraction in the sample favoringJones need not be the same as thefraction of the population whofavor him.If a bal-anced coin is tossed, it is intuitively obvious that the true proportion of times it willturnupheadsis.5.Yetifwesampletheoutcomesforourcoinbytossingit20timestheproportionof heads will varyfrom sampleto sample;that is,on oneoccasionwemightobserve12headsout of 20flips,fora sampleproportionof12/20=.6.On another occasion, we might observe 8 heads out of 20 flips, for a sample pro-portion of 8/20 =.4. In fact, the sample proportion of heads could be 0,.05,.10,.,1.0.Did we conclude that Jones would win because it would be impossible for20 outof 20 sample voters tofavor him if in fact lessthan 50%of the electorate intended tovote for him? The answer to this question is certainly no, but it provides the key toourhidden lineof logic.It isnot impossibletodraw20out of 20favoring Joneswhenlessthan50%oftheelectoratefavorhim,but itishighlyimprobable.Asaresult,weconcluded that hewould win.Copyright 2011 CengaRiaeR
1.4 How Inferences Are Made 13 1.20 Weekly maintenance costs for a factory, recorded over a long period of time and adjusted for inflation, tend to have an approximately normal distribution with an average of $420 and a standard deviation of $30. If $450 is budgeted for next week, what is an approximate probability that this budgeted figure will be exceeded? 1.21 The manufacturer of a new food additive for beef cattle claims that 80% of the animals fed a diet including this additive should have monthly weight gains in excess of 20 pounds. A large sample of measurements on weight gains for cattle fed this diet exhibits an approximately normal distribution with mean 22 pounds and standard deviation 2 pounds. Do you think the sample information contradicts the manufacturer’s claim? (Calculate the probability of a weight gain exceeding 20 pounds.) 1.4 How Inferences Are Made The mechanism instrumental in making inferences can be well illustrated by analyzing our own intuitive inference-making procedures. Suppose that two candidates are running for a public office in our community and that we wish to determine whether our candidate, Jones, is favored to win. The population of interest is the set of responses from all eligible voters who will vote on election day, and we wish to determine whether the fraction favoring Jones exceeds .5. For the sake of simplicity, suppose that all eligible voters will go to the polls and that we randomly select a sample of 20 from the courthouse roster of voters. All 20 are contacted and all favor Jones. What do you conclude about Jones’s prospects for winning the election? There is little doubt that most of us would immediately infer that Jones will win. This is an easy inference to make, but this inference itself is not our immediate goal. Rather, we wish to examine the mental processes that were employed in reaching this conclusion about the prospective behavior of a large voting population based on a sample of only 20 people. Winning means acquiring more than 50% of the votes. Did we conclude that Jones would win because we thought that the fraction favoring Jones in the sample was identical to the fraction favoring Jones in the population? We know that this is probably not true. A simple experiment will verify that the fraction in the sample favoring Jones need not be the same as the fraction of the population who favor him. If a balanced coin is tossed, it is intuitively obvious that the true proportion of times it will turn up heads is .5. Yet if we sample the outcomes for our coin by tossing it 20 times, the proportion of heads will vary from sample to sample; that is, on one occasion we might observe 12 heads out of 20 flips, for a sample proportion of 12/20 = .6. On another occasion, we might observe 8 heads out of 20 flips, for a sample proportion of 8/20 = .4. In fact, the sample proportion of heads could be 0, .05, .10, ., 1.0. Did we conclude that Jones would win because it would be impossible for 20 out of 20 sample voters to favor him if in fact less than 50% of the electorate intended to vote for him? The answer to this question is certainly no, but it provides the key to our hidden line of logic. It is not impossible to draw 20 out of 20 favoring Jones when less than 50% of the electorate favor him, but it is highly improbable. As a result, we concluded that he would win. Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it
14Chapter 1What Is Statistics?Thisexample illustratesthepotentroleplayed byprobabilityinmakinginferences.Probabilists assumethat theyknowthe structure of the population ofinterest andusethetheoryof probabilityto computethe probability ofobtaining a particular sample.Assuming that they know the structure of a population generated by random drawingsoffivecardsfrom a standard deck,probabilistscompute theprobability thatthedrawwill yield three aces and two kings. Statisticians use probability to make the trip inreversefrom the sample to the population. Observing five aces in a sample of fivecards, they immediately infer that the deck (which generates the population) is loadedand not standard. The probability of drawing five aces from a standard deck is zero!This is an exaggerated case,but it makes the point. Basic to inference making is theproblem of calculating the probabilityof an observed sample.As a result,probabilityis themechanismusedin making statistical inferences.One final comment is in order. If you did not think that the sample justified aninference that Jones would win, do not feel too chagrined.One can easily bemisledwhen making intuitive evaluations of the probabilities of events.If you decided thattheprobabilitywas verylowthat20votersoutof 20 wouldfavor Jones,assumingthatJones would lose,you were correct.However, it is not difficult to concoctan examplein whichan intuitive assessmentof probability would be in error.Intuitiveassessmentsof probabilities are unsatisfactory, and we need a rigorous theory of probability inorder to develop methods of inference.1.5Theory and Reality1Theories are conjectures proposed to explain phenomena in the real world. As such,theories are approximations or models for reality.These models or explanations ofreality are presentedin verbal form in some less quantitative fields and as mathematicalrelationships in others.Whereas a theory of social changemightbe expressed verballyin sociology,a description of the motion ofa vibrating string is presented in a precisemathematical manner in physics. When we choose a mathematical model for a phys-ical process,wehopethatthemodel reflectsfaithfully,in mathematical terms,theattributes of thephysical process.If so,the mathematical model can be used to arriveat conclusions about the process itself. If we could develop an equation to predict theposition of a vibrating string,the quality of the prediction would depend on how wellthe equation fit the motion of the string.The process of finding a good equation isnot necessarily simpleandusuallyrequires several simplifyingassumptions (uniformstring mass, no air resistance, etc.).The final criterion for deciding whether a modelis “"good" is whether it yields good and useful information. The motivation for usingmathematical models lies primarily in their utility.This text is concerned with the theory of statistics and hence with models of reality.We will postulate theoretical frequency distributions for populations and will developa theory of probability and inference in a precise mathematical manner.The net resultwill be a theoretical or mathematical model for acquiring and utilizing informationin real life.The model will not be an exact representation of nature, but this shouldnot disturb us. Its utility, like that of other theories, will be measured by its ability toassist us in understanding nature and in solving problems in the real world.Copyright 2011 Cengage LearninAllRightscahChapter(s)Editorial
14 Chapter 1 What Is Statistics? This example illustrates the potent role played by probability in making inferences. Probabilists assume that they know the structure of the population of interest and use the theory of probability to compute the probability of obtaining a particular sample. Assuming that they know the structure of a population generated by random drawings of five cards from a standard deck, probabilists compute the probability that the draw will yield three aces and two kings. Statisticians use probability to make the trip in reverse—from the sample to the population. Observing five aces in a sample of five cards, they immediately infer that the deck (which generates the population) is loaded and not standard. The probability of drawing five aces from a standard deck is zero! This is an exaggerated case, but it makes the point. Basic to inference making is the problem of calculating the probability of an observed sample. As a result, probability is the mechanism used in making statistical inferences. One final comment is in order. If you did not think that the sample justified an inference that Jones would win, do not feel too chagrined. One can easily be misled when making intuitive evaluations of the probabilities of events. If you decided that the probability was very low that 20 voters out of 20 would favor Jones, assuming that Jones would lose, you were correct. However, it is not difficult to concoct an example in which an intuitive assessment of probability would be in error. Intuitive assessments of probabilities are unsatisfactory, and we need a rigorous theory of probability in order to develop methods of inference. 1.5 Theory and Reality Theories are conjectures proposed to explain phenomena in the real world. As such, theories are approximations or models for reality. These models or explanations of reality are presented in verbal form in some less quantitative fields and as mathematical relationships in others. Whereas a theory of social change might be expressed verbally in sociology, a description of the motion of a vibrating string is presented in a precise mathematical manner in physics. When we choose a mathematical model for a physical process, we hope that the model reflects faithfully, in mathematical terms, the attributes of the physical process. If so, the mathematical model can be used to arrive at conclusions about the process itself. If we could develop an equation to predict the position of a vibrating string, the quality of the prediction would depend on how well the equation fit the motion of the string. The process of finding a good equation is not necessarily simple and usually requires several simplifying assumptions (uniform string mass, no air resistance, etc.). The final criterion for deciding whether a model is “good” is whether it yields good and useful information. The motivation for using mathematical models lies primarily in their utility. This text is concerned with the theory of statistics and hence with models of reality. We will postulate theoretical frequency distributions for populations and will develop a theory of probability and inference in a precise mathematical manner. The net result will be a theoretical or mathematical model for acquiring and utilizing information in real life. The model will not be an exact representation of nature, but this should not disturb us. Its utility, like that of other theories, will be measured by its ability to assist us in understanding nature and in solving problems in the real world. Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it
ReferencesandFurtherReadings151.6 SummaryThe objective of statistics is to make an inference about a population based on infor-mation contained in a sample taken from that population.The theory of statistics isa theory of information concerned with quantifying information, designing experi-ments or procedures for data collection, and analyzing data.Our goal is to minimizethe cost of a specified quantity of information and to use this information to make in-ferences.Most important, wehave viewed making an inference about the unknownpopulation as a two-step procedure.First, we enlist a suitable inferential procedureforthe given situation.Second, we seek a measure of the goodness of theresultinginference.For example,everyestimateofapopulation characteristic based on infor-mation contained in the sample might have associated with it a probabilistic boundon the error of estimation.A necessary prelude to making inferences about a population is the ability to describe a setof numbers.Frequency distributions provide a graphic and useful methodforcharacterizingconceptualorrealpopulations of numbers.Numericaldescriptivemeasures areoftenmoreusefulwhenwewishtomakeaninferenceandmeasurethegoodness of that inference.Themechanismfor making inferences is provided by the theory of probability.Theprobabilistreasons fromaknownpopulationtotheoutcomeofa singleexperiment,the sample.In contrast, the statistician utilizes the theory of probability to calculatethe probabilityof an observed sample and to inferfrom this the characteristics of anunknown population.Thus,probability is the foundation of the theory of statistics.Finally, we have noted the difference between theory and reality. In this text, wewill study the mathematical theory of statistics,which is an idealization of nature.Itisrigorous.mathematical,andsubjecttostudyinavacuumcompletelyisolatedfromthe real world. Or it can be tied very closely to reality and can be useful in makinginferencesfromdatain all fieldsofscience.Inthistext,wewillbeutilitarian.Wewillnotregard statistics as abranch of mathematics butasan area of scienceconcernedwith developing a practical theory of information.We will consider statistics as aseparate field,analogous tophysics-not as a branchof mathematics but as a theoryof information that utilizes mathematics heavily.Subsequent chapters will expand on the topics that we have encountered in thisintroduction. We will begin with a study of the mechanism employed in makinginferences, the theory of probability.This theory provides theoretical models forgenerating experimental data and therebyprovides thebasis for ourstudyofstatisticalinference.ReferencesandFurtherReadingsCleveland, W.S.1994.TheElements of GraphingData.Murray Hill, N.J.:AT&TBell Laboratories..VisualizingData.1993.Summit,N.J.:Hobart Press.Fraser,D.A. S.1958.Statistics, an Introduction.New York:Wiley.Copyright 2011 Ceng
References and Further Readings 15 1.6 Summary The objective of statistics is to make an inference about a population based on information contained in a sample taken from that population. The theory of statistics is a theory of information concerned with quantifying information, designing experiments or procedures for data collection, and analyzing data. Our goal is to minimize the cost of a specified quantity of information and to use this information to make inferences. Most important, we have viewed making an inference about the unknown population as a two-step procedure. First, we enlist a suitable inferential procedure for the given situation. Second, we seek a measure of the goodness of the resulting inference. For example, every estimate of a population characteristic based on information contained in the sample might have associated with it a probabilistic bound on the error of estimation. A necessary prelude to making inferences about a population is the ability to describe a set of numbers. Frequency distributions provide a graphic and useful method for characterizing conceptual or real populations of numbers. Numerical descriptive measures are often more useful when we wish to make an inference and measure the goodness of that inference. The mechanism for making inferences is provided by the theory of probability. The probabilist reasons from a known population to the outcome of a single experiment, the sample. In contrast, the statistician utilizes the theory of probability to calculate the probability of an observed sample and to infer from this the characteristics of an unknown population. Thus, probability is the foundation of the theory of statistics. Finally, we have noted the difference between theory and reality. In this text, we will study the mathematical theory of statistics, which is an idealization of nature. It is rigorous, mathematical, and subject to study in a vacuum completely isolated from the real world. Or it can be tied very closely to reality and can be useful in making inferences from data in all fields of science. In this text, we will be utilitarian. We will not regard statistics as a branch of mathematics but as an area of science concerned with developing a practical theory of information. We will consider statistics as a separate field, analogous to physics—not as a branch of mathematics but as a theory of information that utilizes mathematics heavily. Subsequent chapters will expand on the topics that we have encountered in this introduction. We will begin with a study of the mechanism employed in making inferences, the theory of probability. This theory provides theoretical models for generating experimental data and thereby provides the basis for our study of statistical inference. References and Further Readings Cleveland, W. S. 1994. The Elements of Graphing Data. Murray Hill, N.J.: AT&T Bell Laboratories. ———. Visualizing Data. 1993. Summit, N.J.: Hobart Press. Fraser, D. A. S. 1958. Statistics, an Introduction. New York: Wiley. Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it
16Chapter1What Is Statistics?Freund,J.E.,andR.E.Walpole.1987.Mathematical Statistics,4thed.EnglewoodCliffs, N.J.:PrenticeHall.Iman, R.L.1994.A Data-Based Approach to Statistics.Belmont, Calif.: DuxburyPress.Mendenhall, W.,R.J.Beaver, and B.M.Beaver.2006.Introduction toProbabilityandStatistics,12thed.Belmont,Calif.:DuxburyPress.Mood, A.M.,F.A.Graybill, and D.Boes.1974.Introduction to theTheory ofStatistics,3rded.New York:McGraw-Hill.Moore, D.S.,and G.P.McCabe.2002. Introduction to the Practice of Statistics,4thed.NewYork:Freeman.Rice,J.A.Mathematical Statistics and Data Analysis,2nd ed.Belmont, Calif.:DuxburyPress,1995.Stuart, A., and J.K. Ord. 1991.Kendall's Theory of Statistics, 5th ed., vol.1.London:EdwardArnold.SupplementaryExercises1.22Prove that the sum of the deviations ofa set of measurements about their mean is equal to zero;that is,(i -) =0.1.23The mean duration of television commercials is 75 seconds with standard deviation 20 seconds.Assume that the durations are approximately normally distributed to answer the followingaWhatpercentage of commercialslast longerthan95seconds?bWhat percentage of the commercials last between 35 and 115 seconds?CWould you expect commercial to last longer than 2 minutes? Why or why not?1.24Aqua running has been suggested as a method of cardiovascular conditioning for injuredathletes and others who desire a low-impact aerobics program. In a study to investigate therelationship between exercise cadence and heart rate, the heart rates of 20 healthy volunteersweremeasured atacadenceof 48cyclesperminute(acycleconsisted of twosteps).Thedataare as follows:87109798096959092969810191781129498941078196Use the range of the measurements to obtain an estimate of the standard deviation.abConstruct a frequency histogramfor thedata.Use the histogram to obtain a visual approx-imation to y and s.Calculate y and s. Compare these results with the calculation checks provided by parts (a)and (b).dConstruct the intervals y±ks,k=1,2,and 3,and count thenumber of measurementsfalling in each interval. Compare the fractions falling in the intervals with the fractions thatyouwouldexpectaccordingtotheempirical rule.1. R. P. Wilder, D. Breenan, and D. E. Schotte,"A Standard Measure for Exercise Prescription for AquaRunning"American Jourmal of SportsMedicine21(1) (1993): 45.Copyright 2011 CengaRia
16 Chapter 1 What Is Statistics? Freund, J. E., and R. E. Walpole. 1987. Mathematical Statistics, 4th ed. Englewood Cliffs, N.J.: Prentice Hall. Iman, R. L. 1994. A Data-Based Approach to Statistics. Belmont, Calif.: Duxbury Press. Mendenhall, W., R. J. Beaver, and B. M. Beaver. 2006. Introduction to Probability and Statistics, 12th ed. Belmont, Calif.: Duxbury Press. Mood, A. M., F. A. Graybill, and D. Boes. 1974. Introduction to the Theory of Statistics, 3rd ed. New York: McGraw-Hill. Moore, D. S., and G. P. McCabe. 2002. Introduction to the Practice of Statistics, 4th ed. New York: Freeman. Rice, J. A. Mathematical Statistics and Data Analysis, 2nd ed. Belmont, Calif.: Duxbury Press, 1995. Stuart, A., and J. K. Ord. 1991. Kendall’s Theory of Statistics, 5th ed., vol. 1. London: Edward Arnold. Supplementary Exercises 1.22 Prove that the sum of the deviations of a set of measurements about their mean is equal to zero; that is, bn i=1 (yi − y) = 0. 1.23 The mean duration of television commercials is 75 seconds with standard deviation 20 seconds. Assume that the durations are approximately normally distributed to answer the following. a What percentage of commercials last longer than 95 seconds? b What percentage of the commercials last between 35 and 115 seconds? c Would you expect commercial to last longer than 2 minutes? Why or why not? 1.24 Aqua running has been suggested as a method of cardiovascular conditioning for injured athletes and others who desire a low-impact aerobics program. In a study to investigate the relationship between exercise cadence and heart rate,1 the heart rates of 20 healthy volunteers were measured at a cadence of 48 cycles per minute (a cycle consisted of two steps). The data are as follows: 87 109 79 80 96 95 90 92 96 98 101 91 78 112 94 98 94 107 81 96 a Use the range of the measurements to obtain an estimate of the standard deviation. b Construct a frequency histogram for the data. Use the histogram to obtain a visual approximation to y and s. c Calculate y and s. Compare these results with the calculation checks provided by parts (a) and (b). d Construct the intervals y ± ks, k = 1, 2, and 3, and count the number of measurements falling in each interval. Compare the fractions falling in the intervals with the fractions that you would expect according to the empirical rule. 1. R. P. Wilder, D. Breenan, and D. E. Schotte,“A Standard Measure for Exercise Prescription for Aqua Running,” American Journal of Sports Medicine 21(1) (1993): 45. Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it