Introduction To Statistics & Data Analysis - Wylie

Chapter 1: The Role of Statistics and the Data Analysis Process1.231.245aThe dotplot shows that there were two sites that received far greater numbers of visits thanthe remaining 23 sites. Also, it shows that the distribution of the number of visits has thegreatest density of points for the smaller numbers of visits, with the density decreasing as thenumber of visits increases. This is the case even when only the 23 less popular sites areconsidered.bAgain, it is clear from the dotplot that there were two sites that were used by far greaternumbers of individuals (unique visitors) than the remaining 23 sites. However, these two sitesare less far above the others in terms of the number of unique visitors than they are in termsof the total number of visits. As with the distribution of the total number of visits, thedistribution of the number of unique visitors has the greatest density of points for the smallernumbers of visitors, with the density decreasing as the number of unique visitors increases.This is the case even when only the 23 less popular sites are considered.cThe statistic “visits per unique visitor” tells us how heavily the individuals are using the sites.Although the table tells us that the most popular site (Facebook) in terms of the other twostatistics also has the highest value of this statistic, the dotplot of visits per unique visitorshows that no one or two individual sites are far ahead of the rest in this respect.aIt would not be appropriate to use a dotplot because rating is a categorical variable.bWet Weather Rating Frequency Relative FrequencyA 40.286A20.143B20.143C20.143D20.143F20.143Relative Frequency0.300.250.200.150.100.050.00A ABCWet Weather RatingDF

6Chapter 1: The Role of Statistics and the Data Analysis ProcesscDry Weather Rating Frequency Relative FrequencyA 10.071A90.643B30.214F10.071Relative Frequency0.70.60.50.40.30.20.10.0d1.25A ABDry Weather RatingFYes. Apart the greater proportion of “A ” ratings for wet weather than for dry weather, thebeaches on the whole receive higher ratings in dry weather than in wet weather, with only28.6% of beaches receiving below an A in dry weather, compared to 57.1% in wet weather.aEMW5b1.261015Wireless %2025Looking at the dotplot we can see that Eastern states have, on average, lower wirelesspercents than states in the other two regions. The West and Middle states regions have, onaverage, roughly equal wireless percents.a05101520Number of Violent Crimes2530

Chapter 1: The Role of Statistics and the Data Analysis Process7Five schools seem to stand out from the rest, these being, in increasing order of number ofcrimes, Florida International, Florida A&M, University of Florida, University of CentralFlorida, and Florida State University.bViolent Crime RatePer 1000 StudentsEdison State College0.234Florida A&M University1.060Florida Atlantic University0.158Florida Gulf Coast University0.233Florida International University0.202Florida State University0.755New College of Florida1.183Pensacola State College0.260Santa Fe College0.065Tallahassee Community College0.133University of Central Florida0.445University of Florida0.363University of North Florida0.123University of South Florida0.464University of West 0.961.12Violent Crimes per 1000 StudentsThe colleges that stand out in violent crimes per 1000 students are, in increasing order ofcrime rate, Florida State University, Florida A&M University, and New College of Florida.Only Florida A&M stands out in both boxplots.1.27cFor the number of violent crimes, there are five schools that stand out by having highnumbers of crimes, with the majority of the schools having similar, and low, numbers ofcrimes. There seems to be greater consistency for crime rate (per 1000 students) among the15 schools than there is for number of crimes, with just three schools standing out as havinghigh crime rates, and no schools with crime rates that stand out as being low.aWhen ranking the airlines according to delayed flights, one airline would be ranked aboveanother if the probability of a randomly chosen flight being delayed is smaller for the firstairline than it is for the second airline. These probabilities are estimated using the rate per10,000 flights values, and so these are the data that should be used for this ranking. (Note thatthe total number of flights values are not suitable for this ranking. Suppose that one airlinehad a larger number of delayed flights than another airline. It is possible that this could beaccounted for merely through the first airline having more flights than the second.)bThere are two airlines, ExpressJet and Continental, which, with 4.9 and 4.1 of every 10,000flights delayed, stand out as the worst airlines in this regard. There are two further airlinesthat stand out above the rest: Delta and Comair, with rates of 2.8 and 2.7 delayed flights per

8Chapter 1: The Role of Statistics and the Data Analysis Process10,000 flights. All the other airlines have rates below 1.6, with the best rating being forSouthwest, with a rate of only 0.1 delayed flights per 10,000.1.28aRelative Frequency0.70.60.50.40.30.20.10.0None 10,000 10,000- 20,000 20,000Debtb1.29Most public community college graduates have no debt at all, and a debt of 10,000 or lessaccounts for 85% of the graduates. Among the small minority (15%) of the graduates whohave a debt of more than 10,000, only one third (5% of all graduates) have a debt of morethan okokbsst ost os ubuksntMnlioetmm'ooaRrdnCasbf-cheDitfpuOOmWhere Books PurchasedCabBy far the most popular place to buy books is the campus bookstore, with half of the studentsin the sample buying their books from that source. The next most popular sources are onlinebookstores other than the online version of the campus bookstore and off-campus bookstores,with these two sources accounting for around 35% of students. Purchasing mostly eBookswas the least common response.

Chapter 1: The Role of Statistics and the Data Analysis Process1.30Sleepiness at WorkRelative Frequency (%)A few days each month40A few days each week22A daily occurrence7Never a problem31Relative Frequency (%)403020100 A few days each month A few days each week A daily occurrenceSleepiness at WorkNever a problem1.31Type of HouseholdRelative FrequencyNonfamily0.29Married with children0.27Married without children0.29Single parent0.15Relative i edthwicldhinreardriewoit hutcrldhienMType of Household1.32aThe dotplot for Los Angeles County is shown below.ngSilepanret9

10Chapter 1: The Role of Statistics and the Data Analysis Process0612182430Percent Failing (Los Angeles County)3642A typical percent of tests failing for Los Angeles County is around 16. There is one value thatis unusually high (43), with the other values ranging from 2 to 33. There is a greater densityof points toward the lower end of the distribution than toward the upper end.bThe dotplot for the other counties is shown below.0612182430Percent Failing (Other Counties)3642A typical percent of tests failing for the other counties is around 3. There is one extremeresult at the upper end of the distribution (40); the other values range from 0 to 17. Thedensity of points is highest at the left hand end of the distribution and decreases as the percentfailing values increase.cThe typical value for Los Angeles County (around 16) is greater than for the other counties(around 3) and, disregarding the one extreme value in each case, there is a greater variabilityin the values for Los Angeles County than for the other counties. In the distribution for LosAngeles County the points are closer to being uniformly distributed than in the distributionfor the other counties, where there is a clear tail-off of density of points as you move to theright of the distribution.

Chapter 1: The Role of Statistics and the Data Analysis Process1.33aCategoricalbRelative Frequency (%)50403020100cBelow BasicBasicIntermediateLiteracy LevelProficientNo, since dotplots are used for numerical data.1.34Frequency140120100806040200Strongly disagreeDisagreeNot sureResponseAgreeStrongly agree11

121.35Chapter 1: The Role of Statistics and the Data Analysis ProcessaRelative Frequency0.40.30.20.10.0yritcueSntea inMceanhtigFlopioaternsdoarazuslsri ateamOtrheHType of Violationb1.36By far the most frequently occurring violation categories were security (43%) andmaintenance (39%). The least frequently occurring violation categories were flight operations(6%) and hazardous materials (3%).a15b20253035Acceptance Rate (%)404550A typical acceptance rate for these top 25 schools is around 30, with the great majority ofacceptance rates being between 19 and 39. There are no particularly extreme values. Thepattern of the points is roughly symmetrical.

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Chapter 10Hypothesis Testing Using a Single SampleNote: In this chapter, numerical answers to questions involving the normal and t distributions were foundusing values from a calculator. Students using statistical tables will find that their answers differ slightlyfrom those given.10.1Legitimate hypotheses concern population characteristics; x is a sample statistic.10.2aDoes not comply. The alternative hypothesis must involve an inequality.bDoes not comply. The inequality in the alternative hypothesis must refer to the hypothesizedvalue.cDoes comply.dDoes not comply. The alternative hypothesis must involve an inequality referring to thehypothesized value.eDoes not comply, since p̂ is not a population characteristic.10.3Because so much is at stake at a nuclear power plant, the inspection team needs to obtainconvincing evidence that everything is in order. To put this another way, the team needs not onlyto obtain a sample mean greater than 100 but, beyond that, to be sure that sample mean issufficiently far above 100 to provide convincing evidence that the true mean weld strength isgreater than 100. Hence an alternative hypothesis of H a : µ 100 will be used.10.4aThe conclusion is consistent with testingH0: concealed weapons laws do not reduce crimeversusHa: concealed weapons laws reduce crime.The hypothesis that concealed weapons do not reduce crime is equivalent to the statementthat the crime rate when the laws are in place is equal to the crime rate when the laws are notin place, and therefore is essentially an equality. Thus this statement is suitable as the nullhypothesis. The hypothesis that concealed weapons reduce crime is equivalent to thestatement that the crime rate when the laws are in place is less than the crime rate when thelaws are not in place, and therefore is essentially an inequality. Thus this statement is suitableas the alternative hypothesis.b10.5The null hypothesis was not rejected, since no evidence was found that the laws werereducing crime.We are clearly talking here about a situation where, in a sample of children who had received theMMR vaccine, a higher incidence of autism was observed than the incidence of autism inchildren in general. The process of the hypothesis test is then to assume that the incidence ofautism is the same amongst the population of children who have had the MMR vaccine as it isamongst children in general, and then to find out whether, on that basis, a result such as the one261

262Chapter 10: Hypothesis Testing Using a Single Sampleobtained in the sample would be very unusual, or not particularly unusual. If such a result wouldbe very unusual, then the sample result is providing convincing evidence of a higher incidence ofautism amongst the population of children who have received the MMR vaccine than in childrenin general. If the sample result would not be particularly unusual, then it would not provideconvincing evidence of this. However, since the incidence of autism amongst children in thesample was observed to be higher than it is known to be in children in general, there’s no waythat this result can provide evidence that MMR does not cause autism.10.6H0: p 1/3 versus Ha: p 1/3, where p is the proportion of employers who have sent an employeehome to change clothes10.7H0: p 0.1 versus Ha: p 0.110.8H0: µ 170 versus Ha: µ 170.10.9Let p be the proportion of all constituents who favor spending money for the new sewer system.She should test H0: p 0.5 versus Ha: p 0.5.10.10H0: μ 10.3 versus Ha: μ 10.310.11H0: p 0.83 versus Ha: p 0.8310.12aType IbA Type I error is coming to the conclusion that cancer is present when, in fact, it is not.Treatment may be started when, in fact, no treatment is necessary.cA Type II error is coming to the conclusion that no cancer is present when, in fact, the illnessis present. No treatment will be prescribed when, in fact, treatment is necessary.dIn terms of adjustment of α levels, decreasing the probability of a Type II error involvesincreasing the probability of a Type I error.aThis is a Type I error. Its probability is 3 33 0.091.bA Type II error would be coming to the conclusion that the woman has cancer in the otherbreast when in fact she does not have cancer in the other breast. The probability that thishappens is 91 936 0.097.aA Type I error is coming to the conclusion that the symptoms are due to disease when in factthe symptoms are due to child abuse.A Type II error is coming to the conclusion that the symptoms are due to child abuse when infact the symptoms are due to disease.bThe doctor considers the presence of child abuse more serious than the presence of disease.Thus, according to the doctor, undetected child abuse is more serious than undetected disease,and a Type I error is the more serious.10.1310.14

Chapter 10: Hypothesis Testing Using a Single Sample10.1510.1610.1710.18263aA Type I error would be coming to the conclusion that the man is not the father when in facthe is. A Type II error would be not coming to the conclusion that the man is not the fatherwhen in fact he is not the father.bα 0.001, β 0.cA “false positive” is coming to the conclusion that the man is the father when in fact he is notthe father. This is a Type II error, and its probability is β 0.008 .aA Type I error would be obtaining convincing evidence that less than 90% of the TV setsneed no repair when in fact (at least) 90% need no repair. The consumer agency might takeaction against the manufacturer when in fact the manufacturer is not at fault.A Type II error would be not obtaining convincing evidence that less than 90% of the TV setsneed no repair when in fact less than 90% need no repair. The consumer agency would nottake action against the manufacturer when in fact the manufacturer is making untrue claimsabout the reliability of the TV sets.bTaking action against the manufacturer when in fact the manufacturer is not at fault couldinvolve large and unnecessary legal costs to the consumer agency. Thus a Type I error couldbe considered serious, whereas a Type II error would only involve not catching amanufacturer who is making false claims. Therefore, in order to reduce the probability of aType I error, a procedure using α 0.01 should be recommended.aA Type I error is obtaining convincing evidence that more than 1% of a shipment is defectivewhen in fact (at least) 1% of the shipment is defective. A Type II error is not obtainingconvincing evidence that more than 1% of a shipment is defective when in fact more than 1%of the shipment is defective.bThe consequence of a Type I error would be that the calculator manufacturer returns ashipment when in fact it was acceptable. This will do minimal harm to the calculatormanufacturer’s business. However, the consequence of a Type II error would be that thecalculator manufacturer would go ahead and use in the calculators circuits that are defective.This will then lead to faulty calculators and would therefore be harmful to the manufacturer’sbusiness. A Type II error would be the more serious for the calculator manufacturer.cAt least in the short term, a Type II error would not be harmful to the supplier’s business;payment would be received for a shipment that was in fact faulty. However, if a Type I errorwere to occur, the supplier would receive back, and not be paid for, a shipment of circuits thatwas in fact acceptable. A Type I error would be the more serious for the supplier.aA Type I error is obtaining convincing evidence that the mean water temperature is greaterthan 150 F when in fact it is (at most) 150 F.A Type II error is not obtaining convincing evidence that the mean water temperature isgreater than 150 F when in fact it is greater than 150 F.bIf a Type II error occurs, then the ecosystem will be harmed and no action will be taken. Thiscould be considered more serious than a Type I error, where a company will be required tochange its practices when in fact it is not contravening the regulations. A Type II error ismore serious.

264Chapter 10: Hypothesis Testing Using a Single Sample10.19The probability of a Type I error is equal to the significance level. Here the aim is to reduce theprobability of a Type I error, so a small significance level (such as 0.01) should be used.10.20aThe area will be closed to fishing if the fish are determined to have an unacceptably highmercury content. Thus we should test H0: µ 5 versus Ha: µ 5.bIf a Type II error occurs, then an unacceptably high mercury level will go undetected, andpeople will continue to fish in the area. This could be considered more serious than a Type Ierror, where fishing will be prohibited in an area where the mercury level is in factacceptable. We thus wish to reduce the probability of a Type II error, and therefore asignificance level of 0.1 should be used.aThe researchers failed to reject H0.bIf the researchers were incorrect in their conclusion, then they would be failing to reject H0when H0 was in fact true. This is a Type II error.cYes. The study did not provide convincing evidence that there is a higher cancer death ratefor people who live close to nuclear facilities. However, this does not mean that there was nosuch effect, and this would be the case for any study with the same outcome.aThe conversion will be undertaken only if there is strong evidence that the proportion ofdefective installations is lower for the robots than for human assemblers. Thus themanufacturer should test H0: p 0.02 versus Ha: p 0.02.bA Type I error would be obtaining convincing evidence that the proportion of defectiveinstallations for the robots

1 Chapter 1 The Role of Statistics and the Data Analysis Process 1.1 Descriptive statistics is the branch of statistics that involves the organization and summary of the values in a data set. Inferential statistics is the branch of statistics concerned with reaching conclusions about a population based on the information provided by a sample.