Solution Manual for Essentials of Statistics for the Behavioral Sciences , 2nd Edition

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C-1C H A P T E R11.1Descriptive statistics organize, summarize, and communicate agroup of numerical observations. Inferential statistics use sampledata to make general estimates about the larger population.1.2A sample is a set of observations drawn from the population ofinterest, which we hope shares the same characteristics as thepopulation of interest. A population includes all possibleobservations about which we’d like to know something.1.3The four types of variables are nominal, ordinal, interval, andratio. A nominal variable is used for observations that havecategories, or names, as their values. An ordinal variable is usedfor observations that have rankings (i.e., 1st, 2nd, 3rd . . .) as theirvalues. An interval variable has numbers as its values; the distance(or interval) between pairs of consecutive numbers is assumed tobe equal. Finally, a ratio variable meets the criteria for intervalvariables but also has a meaningful zero point. Interval and ratiovariables are both often referred to as scale variables.1.4Statisticians usescaleas another term for an interval or ratiomeasure. They also use scale as a word for many measurementtools, particularly those that involve a series of items that test-takers must complete.1.5Discrete variables can only be represented by specific numbers,usually whole numbers; continuous variables can take on anyvalues, including those with great decimal precision (e.g.,1.597).1.6An independent variable is a variable that we either manipulateor observe to determine its effects on the dependent variable; adependent variable is the outcome variable that we hypothesizeto be related to, or caused by, changes in the independentvariable.1.7A confounding variable (also called a confound) is any variablethat systematically varies with the independent variable so thatwe cannot logically determine which variable affects thedependent variable. Researchers attempt to controlconfounding variables in experiments by randomly assigningparticipants to conditions. The hope with random assignment isthat the confounding variable will be spread equally across thedifferent conditions of the study, thus neutralizing its effects.1.8Reliability refers to the consistency of a measure. Validity refersto the extent to which a test actually measures what it wasintended to measure. A measure that is valid absolutely must bereliable, but a reliable measure is not necessarily a valid one.1.9An operational definition specifies the operations or proceduresused to measure or manipulate an independent or dependentvariable.1.10In everyday language, people often use the wordexperimenttorefer to something they are trying out to see what will happen.Researchers use the term to refer to a type of study in whichparticipants are randomly assigned to levels of the independentvariable.1.11When conducting experiments, the researcher randomly assignsparticipants to conditions or levels of the independent variable.When random assignment is not possible, such as whenstudying something like gender or marital status, correlationalresearch is used. Correlational research allows us to examinehow variables are related to each other; experimental researchallows us to make assertions about how an independentvariable causes an effect in a dependent variable.1.12In a between-groups research design, participants experienceone, and only one, level of the independent variable. In awithin-groups research design, all participants in the studyexperience all levels of the independent variable.1.13a.“This was an experiment” (not “This was a correlationalstudy”)b.“. . . the independent variable of caffeine . . .” (not “. . . thedependent variable of caffeine. . . .”)c.“A university assessed the validity . . .” (not “A universityassessed the reliability . . .”)d.“In a between-groups experiment . . .” (not “In a within-groups experiment . . .”)1.14a.“. . . the nominal variable ‘gender’ . . .” (not “. . . the ordinalvariable ‘gender’ . . .”)b.“A psychologist used a within-groups design . . .” (not “Apsychologist used a between-groups design . . .”)c.“. . . the effects of the independent variable . . .” (not“. . . the effects of the confounding variable . . .”)d.“A researcher studied a sample of 20 rats . . .” (not “Aresearcher studied a population of 20 rats . . .”)1.15The sample is the 100 customers who completed the survey.The population is all of the customers at the grocery store.1.16a.130 peopleb.All people living in urban areas in the United Statesc.Descriptive statisticAPPENDIXCSolutions to End-of-Chapter Problems

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C-2APPENDIX Cd.Answers may vary, but one way is to sort people intogroups such as “long distance walked,” “medium distancewalked,” and “short distance walked.”e.Answers may vary, but pedometers could be used tomeasure steps taken or miles walked, both of which arescale measures.1.17a.73 peopleb.All people who shop in grocery stores similar to the onewhere data were collectedc.Inferential statisticd.Answers may vary, but people could be labeled as having a“healthy diet” or an “unhealthy diet.”e.Answers may vary, but there could be groupings such as “noitems,” “a minimal number of items,” “some items,” and“many items.”f.Answers may vary, but the number of items could becounted or weighed.1.18Answers may vary, but on a national level, one could look atthe rate of houses in foreclosure or the amount of governmentdebt.1.19a.The independent variables are physical distance andemotional distance. The dependent variable is accuracy ofmemory.b.There are two levels of physical distance (within 100 milesand 100 miles or farther) and three levels of emotionaldistance (knowing no one who was affected, knowingpeople who were affected but lived, and knowing someonewho died).c.Answers may vary, but accuracy of memory could beoperationalized as the number of facts correctly recalled.1.20a.Skin toneb.Severity of facial wrinklesc.Three levels (light, medium, and dark)1.21a.The average weight for a 10-year-old girl was 77.4 poundsin 1963 and nearly 88 pounds in 2002.b.No; the CDC would not be able to weigh every single girlin the United States because it would be too expensive andtime consuming.c.It is a descriptive statistic because it is a numerical summaryof a sample. It is an inferential statistic because theresearchers drew conclusions about the population’s averageweight based on this information from a sample.1.22a.The sample is the 60,000 people they studied.b.The researchers would like to generalize their findings tothe population of all Norwegians, or perhaps even morebroadly.1.23a.Ordinalb.Scalec.Nominal1.24a.Ordinalb.Scalec.Scaled.Nominale.Nominal1.25a.Discreteb.Continuousc.Discreted.Discretee.Continuous1.26a.A reliable test is one that provides consistent results. If youtake the test twice, you should get the same results, anindication of reliability.b.A valid test is one that measures what it intends to measure.This test has the stated intention of measuring personality. Ifin fact it is measuring personality accurately, then it is avalid test.c.There are several possible answers to this question. Thedevelopers of this Web site might, for example, hypothesizethat the region of the world in which one grew up predictsdifferent personality profiles that are based on region.d.The independent variable would be region and thedependent variable would be personality profile.1.27a.The independent variables are temperature and rainfall.Both are continuous, scale variables.b.The dependent variable is experts’ ratings. This is a discrete,scale variable.c.The researchers wanted to know if the wine experts areconsistent in their ratings—that is, if they’re reliable.d.This observation would suggest that Robert Parker’sjudgments are valid. His ratings seem to be measuring whatthey intend to measure—wine quality.1.28a.(1) Measured the distance between the well and the homeson a map; (2) measuring how many steps it takes to walkfrom a home to the wellb.(1) Described hair as short, medium, or very long;(2) measuring the length of the hair in inches1.29a.Age: teenagers and adults in their 30s; video gameperformance: final score on a video game or averagereaction time on a video game taskb.Spanking: spanking and not spanking; violent behavior:parental measure of child aggression or number ofaggressive acts observed in an hour of playc.Meetings: go to meetings and participate online; weight loss:measured in pounds or kilograms, or by change in waist sized.Studying: with others and alone; statistics performance:average test score for the semester or overall grade for thesemestere.Beverage: caffeinated and decaffeinated; time to fall asleep:minutes to fall asleep from when the participant goes tobed, or the actual time at which the participant falls asleep1.30a.The study could use a between-groups research design byassigning half the participants to exercise and half not toexercise.b.Participants could be followed for several months todetermine weight loss before the exercise program, thenstart the exercise program and be followed for severalmonths to determine weight loss after the program.c.There are several possible confounds. In the within-groupsdesign, the participants are having their weight loss tracked,then starting an exercise program, then having their weightloss tracked some more. It is possible that the mere act of

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APPENDIX CC-3tracking weight loss leads participants to implement weight-loss tactics other than exercise and that they start reapingthe benefits of these tactics around the time the exerciseprogram begins. Alternatively, it is possible that the no-exercise segment occurs in the winter and the exercisesegment occurs in the spring. Many people gain a bit ofweight during the winter and lose weight as summer—andbathing-suit season—approaches. It might be the weather,not the exercise program, that leads to weight loss.1.31a.An experiment requires random assignment to conditions.It would not be ethical to randomly assign some people tosmoke and some people not to smoke, so this research hadto be correlational.b.Other unhealthy behaviors have been associated withsmoking, such as poor diet and infrequent exercise. Theseother unhealthy behaviors might be confounded withsmoking.c.The tobacco industry could claim it was not the smokingthat was harming people, but rather the other activities inwhich smokers tend to engage or fail to engage.d.You could randomly assign people to either a smokinggroup or a nonsmoking group, and assess their health overtime.1.32a.This research is correlational because participants could notbe randomly assigned to be high in individualism orcollectivism.b.The sample is the 32 people who tested high forindividualism and the 37 people who tested high forcollectivism.c.Answers may vary, but one hypothesis could be “Onaverage, people high in individualism will have morerelationship conflict than those high in collectivism.”d.Answers may vary, but one way to measure relationshipconflict could be counting the number of disagreements orfights per month.1.33a.This is experimental because students are randomly assignedto one of the incentive conditions for recycling.b.Answers may vary, but one hypothesis could be “Studentsfined for not recycling will report lower concerns for theenvironment, on average, than those rewarded forrecycling.”1.34a.Participants in the Millennium Cohort Study.b.Parents in the United Kingdom, or possibly all parentsglobally.c.This is a correlational study, as individuals were notrandomly assigned to the condition of being a marriedcouple or a cohabitating couple.d.Marital status—married or cohabitinge.Length of relationshipf.There are several possible answers to this question. Forexample, economic status or financial well-being may be aconfounding factor, as those who are more likely to havethe money to marry and raise a family may have fewer lifestressors than those who have less money, do not marry, andchoose to cohabitate. This variable could be operationalizedand measured via household income.1.35a.Researchers could have randomly assigned some peoplewho are HIV-positive to take the oral vaccine and otherpeople who are HIV-positive not to take the oral vaccine.The second group would likely take a placebo.b.This would have been a between-groups experimentbecause the people who are HIV-positive would have beenin only one group: either vaccine or no vaccine.c.This limits the researchers’ ability to draw causal conclusionsbecause the participants who received the vaccine may havebeen different in some way from those who did not receivethe vaccine. There may have been a confounding variablethat led to these findings. For example, those who receivedthe vaccine might have had better access to health care andbetter sanitary conditions to begin with, making them lesslikely to contract cholera regardless of the vaccine’seffectiveness.d.The researchers might not have used random assignmentbecause it would have meant recruiting participants, likelyimmunizing half, then following up with all of them. Theresearchers likely did not want to deny the vaccine topeople who were HIV-positive because they might havecontracted cholera and died without it.e.We could have recruited a sample of people who wereHIV-positive. Half would have been randomly assigned totake the oral vaccine; half would have been randomlyassigned to take something that appeared to be an oralvaccine but did not have the active ingredient. They wouldhave been followed to determine whether they developedcholera.1.36a.Ability level, graduate level (high school versus university),raceb.Wagesc.12,000 men and women in the United States who were14–22 years old in 1979d.High school and college graduate men and women in theUnited Statese.Participants were studied over a period of time to measurechange during that time period.f.Age could be a confounding variable, as those who areolder will have greater exposure to the various areasmeasured via the AFQT, in addition to the education theyreceived at the college level.g.Ability could be operationalized by having managers rateeach participant’s ability to perform his or her job. Anotherway ability could be operationalized is via high school andcollege GPA or a standardized ability test.C H A P T E R22.1Raw scores are the original data, to which nothing has beendone.2.2To create a frequency table: (1) Determine the highest andlowest scores. (2) Create two columns; label the first with thevariable name and label the second “Frequency.” (3) List thefull range of values that encompasses all the scores in the dataset, from lowest to highest, even those for which the frequencyis 0. (4) Count the number of scores at each value, and writethose numbers in the frequency column.2.3A frequency table is a visual depiction of data that shows howoften each value occurred; that is, it shows how many scoresare at each value. Values are listed in one column, and the

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C-4APPENDIX Cnumbers of individuals with scores at that value are listed in thesecond column. A grouped frequency table is a visual depictionof data that reports the frequency within each given interval,rather than the frequency for each specific value.2.4Statisticians might useintervalto describe a type of variable.Interval variables have numbers as their values, and the distance(or interval) between numbers is assumed to be equal.Statisticians might also useintervalto refer to the range ofvalues to be used in a grouped frequency table, histogram, orpolygon.2.5Bar graphs typically provide scores for nominal data, whereashistograms typically provide frequencies for scale data. Also, thecategories in bar graphs do not need to be arranged in aparticular order and the bars should not touch, whereas theintervals in histograms are arranged in a meaningful order(lowest to highest) and the bars should touch each other.2.6Thex-axis is typically labeled with the name of the variable ofinterest. They-axis is typically labeled “Frequency.”2.7A histogram looks like a bar graph but is usually used to depictscale data, with the values (or midpoints of intervals) of thevariable on thex-axis and the frequencies on they-axis. Afrequency polygon is a line graph, with thex-axis representingvalues (or midpoints of intervals) and they-axis representingfrequencies; a dot is placed at the frequency for each value (ormidpoint), and the points are connected.2.8Visual displays of data often help us see patterns that are notobvious when we examine a long list of numbers. They help usorganize the data in meaningful ways.2.9In everyday conversation, you might use the worddistributionina number of different contexts, from the distribution of food toa marketing distribution. A statistician would usedistributiononly to describe the way that a set of scores, such as a set ofgrades, is distributed. A statistician is looking at the overallpattern of the data—what the shape is, where the data tend tocluster, and how they trail off.2.10A normal distribution is a specific frequency distribution that isa bell-shaped, symmetric, unimodal curve.2.11With positively skewed data, the distribution’s tail extends tothe right, in a positive direction, and with negatively skeweddata, the distribution’s tail extends to the left, in a negativedirection.2.12A floor effect occurs when there are no scores below a certainvalue; a floor effect leads to a positively skewed distributionbecause the lower part of the distribution is constrained.2.13A ceiling effect occurs when there are no scores above acertain value; a ceiling effect leads to a negatively skeweddistribution because the upper part of the distribution isconstrained.2.144.98% and 2.27%2.1517.95% and 40.67%2.163.69% and 18.11% are scale variables, both as counts and aspercentages.2.170.10% and 96.77%2.181,889.00, 2.65, and 0.082.190.04, 198.22, and 17.892.20a.The full range is the maximum (27) minus the minimum(0), plus 1, which equals 28.b.Fivec.The intervals would be 0–4, 5–9, 10–14, 15–19, 20–24, and25–29.2.21The full range of data is 68 minus 2, plus 1, or 67. The range(67) divided by the desired seven intervals gives us an intervalsize of 9.57, or 10 when rounded. The seven intervals are: 0–9,10–19, 20–29, 30–39, 40–49, 50–59, and 60–69.2.2237.5, 52.5, and 67.52.2325 shows2.24Twelve countries had between 2 and 10 first- or second-placeWorld Cup finishes.2.25Serial killers would create positive skew, adding high numbersof murders to the data that are clustered around 1.2.26People convicted of murder are assumed to have killed at leastone person, so observations below one are not seen, whichcreates a floor effect.2.27a.For the college population, the range of ages extends fartherto the right (with a larger number of years) than to the left,creating positive skew.b.The fact that youthful prodigies have limited access tocollege creates a sort of floor effect that makes low scoresless possible.2.28a.Assuming that most people go for the maximum numberof friends, for the range of Facebook friends, thenumber of friends extends farther to the left (with fewernumber of friends) than to the right, creating a negativeskew.b.The fact that Facebook cuts off or limits the number offriends to 5000 means there is a ceiling effect that makeshigher scores impossible.2.29a.b.10.53% of these schools had exactly 4% of their studentsreport that they wrote between 5 and 10 twenty-pagepapers that year.c.This is not a random sample. It includes schools that choseto participate in this survey and opted to have their resultsmade public.PERCENTAGEFREQUENCYPERCENTAGE1015.26900.00800.00700.00600.005210.534210.533421.052421.051526.32015.26

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APPENDIX CC-5d.e.Onef.The data are clustered around 1% to 4%, with a highoutlier, 10%.2.30a.b.30c.A grouped frequency table is not necessary here. These dataare relatively easy to interpret in the frequency table.Grouped frequency tables are useful when the list of data islong and difficult to interpret.d.These data are clustered around 6 to 8 years, with a longtail of data out to a greater number of years to complete.These data show positive skew.e.2143765111210980102345678910 11 1214 1513YearsFrequencyYEARS TO COMPLETEFREQUENCY15214113112111110294897116101236879540 0 13FrequencyPercent of students57911f.Eight2.31a.The variable of alumni giving was operationalized by thepercentage of alumni who donated to a given school. Thereare several other ways it could be operationalized. Forexample, the data might consist of the total dollar amountor the mean dollar amount that each school received.b.c.There are many possible answers to this question. Forexample, we might ask whether sports team success predictsalumni giving or whether the prestige of the institution is afactor (the higher the ranking, the more alumni whodonate).d.e.f.There is one unusual score—61. The distribution appears tobe positively skewed. The center of the distribution seemsto be in the 10–29 range.510152025051525253545556575Percentage of alumni donationsFrequency5101520250515025 35455565Percentage of alumni donationsFrequencyINTERVALFREQUENCY60–69150–59040–49630–391520–292110–19240–93

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C-6APPENDIX C2.32a.b.c.The summary will differ for each student but shouldinclude the following information: the data appear to beroughly symmetric, maybe a bit negatively skewed.d.There are many possible answers to this question. Forexample, one might ask whether teams with older playersdo better or worse than those with younger players.Another study might examine whether team budget relatesto wins; there’s a salary cap, but some teams might chooseto pay the “luxury tax” in order to spend more. Doesspending make a difference?2.33a.Extroversion scores are most likely to have a normaldistribution. Most people would fall toward the middle,with some people having higher levels and some havinglower levels.b.The distribution of finishing times for a marathon is likelyto be positively skewed. The floor is the fastest possibletime, a little over 2 hours; however, some runners take aslong as 6 hours or more. Unfortunately for the very, veryslow but unbelievably dedicated runners, many marathonsshut down the finish line 6 hours after the start of the race.c.The distribution of numbers of meals eaten in a dining hallin a semester on a three-meal-a-day plan is likely to benegatively skewed. The ceiling is three times per day,multiplied by the number of days; most people who chooseto pay for the full plan would eat many of these meals. Afew would hardly ever eat in the dining hall, pulling the tailin a negative direction.2.34a.You would present individual data values because the fewcategories of eye color would result in a readable list.Frequency tableb.You would present grouped data because it is possible foreach person to use a different amount of minutes and such24610805152535455565WinsFrequency97531INTERVALFREQUENCY60–69150–59740–491030–39720–29210–193a long list would be unreadable. Grouped frequency table,histogram, or frequency polygonc.You would present grouped data because time to completecarried out to seconds would produce too many uniquenumbers to organize meaningfully without groupings.Grouped frequency table, histogram, or frequency polygond.You would present individual data values because numberof siblings tends to take on limited values. Frequency table,histogram, or frequency polygon2.35a.b.This is not a random sample because only résumés fromthose applying for a receptionist position in his office wereincluded in the sample.c.This information lets the trainees know that most of theserésumés contained between 220 and 299 words. Thisanalysis tells us nothing about how word count might relateto quality of résumé.2.36a.A histogram of grouped frequenciesb.Approximately 32c.Approximately 27d.Two questions we might ask are (1) How close is theperson to those photographed?, and (2) What mightaccount for the two peaks in these data?e.f.g.The data have two high points around 3–9 and 15–18. Wecan see that the data are asymmetric to the right, creatingpositive skew.246801.54.507.5 10.5 13.5 16.5 19.5Number of people picturedFrequencyINTERVALFREQUENCY18–20215–17612–1429–1136–873–58INTERVALFREQUENCY300–3394260–2997220–2599180–2193

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APPENDIX CC-72.37a.b.c.d.INTERVALFREQUENCY10–14 months25–9 months30–4 months15234516021032145678910 11 12 13MonthsFrequency012345102345678910 11 12 13FrequencyMONTHSFREQUENCYPERCENTAGE1215110010159158007156155004153420221013150525e.f.g.These data are centered around the 3-month period, withpositive skew extending the data out to the 12-monthperiod.h.The bulk of the data would need to be shifted from the 3-month period to approximately 12 months, so that group ofwomen might be the focus of attention. Perhaps earlycontact at the hospital and at follow-up visits after birthwould help encourage mothers to breast-feed, and to breast-feed longer. One could also consider studying the womenwho create the positive skew to learn what uniquecharacteristics or knowledge they have that influenced theirbehavior.2.38a.The column for faculty shows a high point from 0–7friends.b.The column for students shows two high points around 4–11 and 16–23, with some high outliers creating positiveskew.c.The independent variable would be status, with two levels(faculty, student).d.The dependent variable would be number of friends.e.A confounding variable could be age, as faculty are olderthan students and tend to be less involved in social activitiesor situations where making friends is common.f.The dependent variable could be operationalized as thenumber of people who appear in photographs on display234516879111213141015002.522.557.51012.51517.5MonthsFrequency123654089101312151411702.557.512.51015MonthsFrequency

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C-8APPENDIX Cin dorm rooms and offices across campus, as was done forthis study. There are several additional ways these datacould be operationalized. One way would be to record thenumber of Facebook friends each person has. Another waywould be to count the number of friends each personreports interacting with on a regular basis. This lattermethod of measuring number of friends is more likely toreveal the quality of friendship via the amount ofinteraction.2.39a.b.c.This distribution is positively skewed.5101520250Number of students mentored by each different professorFrequency12345678910 11 12141615135101520250304125678101112139Number of studentsFrequencyFORMERSTUDENTS NOWIN TOP JOBSFREQUENCYPERCENTAGE1311.851200.001100.001000.00911.85835.56747.41659.265916.674814.8132342.59d.The researchers operationalized the variable of mentoringsuccess as numbers of students placed into top professorialpositions. There are many other ways this variable couldhave been operationalized. For example, the researchersmight have counted numbers of student publications whilein graduate school or might have asked graduates to ratetheir satisfaction with their graduate mentoringexperiences.e.The students might have attained their professor positionsbecause of the prestige of their advisor, not because of hismentoring.f.There are many possible answers to this question. Forexample, the attainment of a top professor position mightbe predicted by the prestige of the institution, the numberof publications while in graduate school, or the graduatestudent’s academic ability.C H A P T E R33.1The biased scale lie, the sneaky sample lie, the interpolation lie,the extrapolation lie, and the inaccurate values lie.3.2(1) Organize the data by participant; each participant will havetwo scores, one on each scale variable. (2) Label the horizontalx-axis with the name of the independent variable and itspossible values, starting with 0 if practical. (3) Label the verticaly-axis with the name of the dependent variable and its possiblevalues, starting with 0 if practical. (4) Make a mark on thegraph above each study participant’s score on thex-axis andacross from his or her score on they-axis.3.3To convert a scatterplot to a range-frame, simply erase the axesbelow the minimum score and above the maximum score.3.4A linear relation between variables means that the relationbetween variables is best described by a straight line.3.5With scale data, a scatterplot allows for a helpful visual analysisof the relation between two variables. If the data points appearto fall approximately along a straight line, this indicates a linearrelation. If the data form a line that changes direction along itspath, a nonlinear relation may be present. If the data pointsshow no particular relation, it is possible that the two variablesare not related.3.6A line graph is used to illustrate the relation between two scalevariables. One type of line graph is based on a scatterplot andallows us to construct a line of best fit that represents thepredictedyscores for eachxvalue. A second type of line graphallows us to visualize changes in the values on they-axis overtime. A time plot, or time series plot, is a specific type of linegraph. It is a graph that plots a scale variable on they-axis as itchanges over an increment of time (e.g., second, day, century)recorded on thex-axis.3.7A bar graph is a visual depiction of data in which theindependent variable is nominal or ordinal and the dependentvariable is scale. Each bar typically represents the mean value ofthe dependent variable for each category. A Pareto chart is aspecific type of bar graph in which the categories along thex-axis are ordered from highest bar on the left to lowest bar onthe right.

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APPENDIX CC-93.8Bar graphs typically depict summary statistics, such asfrequencies or averages, for several different levels of one ormore nominal or ordinal independent variables. Histogramstypically depict frequencies for different values of one scalevariable. Bars represent counts or percentages for differentvalues of a scale variable or for different intervals of that scalevariable.3.9A pictorial graph is a visual depiction of data typically used fora nominal independent variable with very few levels(categories) and a scale dependent variable. Each level uses apicture or symbol to represent its value on the scale dependentvariable. A pie chart is a graph in the shape of a circle, with aslice for every level. The size of each slice represents theproportion (or percentage) of each category. In most cases, abar graph is preferable to a pictorial graph or a pie chart.3.10Bar graphs are straightforward presentations of data, whereasthe elements of pictorial graphs and pie charts can oftendistract from the data being presented. Also, mistakes inpresentation style are much more common for pictorial graphsand pie charts.3.11The independent variable typically goes on the horizontalx-axis and the dependent variable goes on the verticaly-axis.3.12Whenever possible, graph axes should start at 0, althoughsometimes it is not practical to start at 0. For example, whenthe data do not contain low values (and including 0 wouldminimize the depiction of the actual data), we should use cutmarks to indicate axes that do not start at 0.3.13Moiré vibrations are any visual patterns that create a distractingimpression of vibration and movement. A grid is a backgroundpattern, almost like graph paper, on which the datarepresentations, such as bars, are superimposed. Ducks arefeatures of the data that have been dressed up to be somethingother than merely data.3.14Geographic information systems are particularly powerful foranalyzing demographic patterns or demographic differences ina variable. Knowing how several variables change overgeographic regions could lead researchers to detect importantrelations among variables.3.15Total dollars donated per year is scale data. A time plot wouldnicely show how donations varied across years.3.16Sorting people into the categories of “alumni who donatedmoney” and “alumni who did not donate money” createsnominal data. We would use a bar graph to depict the numbersof alumni who did and did not donate.3.17a.The independent variable is gender and the dependentvariable is video game score.b.Nominalc.Scaled.The best graph for these data would be a bar graph becausethere is a nominal independent variable and a scaledependent variable.3.18Nonlinear, because the data change direction around 4.00 onthex-axis.3.19Linear, because the data could be fit with a line drawn fromthe upper-left to the lower-right corner of the graph.3.20These graphs are missing titles and axis labels. The axes are alsomissing 0 values.3.21a.Bar graphb.Line graph; more specifically, a time plotc.They-axis should go down to 0.d.The lines in the background are grids, and the three-dimensional effect is a type of duck.e.3.20%, 3.22%, 2.80%f.If they-axis started at 0, all of the bars would appear to beabout the same height. The differences would beminimized.3.22These data have a minimum value of 273 and a maximumvalue of 342. Because the minimum value is far from 0, it isnot practical to have the axis start at 0, so cut marks would beused. (However, we would include the full range of data—0to 342—if omitting some of these numbers would bemisleading.) We might then include every 10th value, startingat 270:270, 280, 290, 300, 310, 320, 330, 340, 3503.23The minimum value is 0.04 and the maximum is 0.36, so theaxis could be labeled from 0.00 to 0.40. We might choose tomark every 0.05 value:0.00, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, and 0.403.24a.The highest life expectancy is 82 years. The fertility rateassociated with the highest life expectancy is 0.96.b.Yes, this seems to be a linear relation, with the data fitting aline moving from the upper-left to the lower-right cornerof the graph. As the fertility rate increases, the lifeexpectancy at birth decreases.3.25a.The independent variable is height and the dependentvariable is attractiveness. Both are scale variables.b.The best graph for these data would be a scatterplot (whichalso might include a line of best fit if the relation is linear)because there are two scale variables.c.It would not be practical to start the axis at 0. With the dataclustered from 58 to 71 inches, a 0 start to the axis wouldmean that a large portion of the graph would be empty. Wewould use cut marks to indicate that the axis did notinclude all values from 0 to 58. (However, we wouldinclude the full range of data—0 to 71—if omitting someof these numbers would be misleading.)3.26a.The independent variable is time (i.e., week) and thedependent variable is mean depression level.b.Both variables are scale.c.The best graph for these data would be a time plot becausethe social worker is tracking depression levels over a periodof time (20 weeks).3.27a.The independent variable is country and the dependentvariable is male suicide rate.b.Country is a nominal variable and suicide rate is a scalevariable.c.The best graph for these data would be a Pareto chart.Because there are 20 categories along thex-axis, it is best toarrange them in order from highest to lowest.

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C-10APPENDIX Cd.A time series plot could show year on thex-axis andsuicide rate on they-axis. Each country would berepresented by a different color line.3.28a.b.For the most part, the points on the scatterplot do not seemto indicate any particular relation, whether linear orcurvilinear. Low-mileage days (50 to 70 miles) have somelow-climb and some high-climb days, and mid-mileage days(90 to 110 miles) have some low-climb and some highclimb days. Only the two very long mileage days (around120 miles) have low climbs, perhaps indicating a tinyrelation.c.The cyclists experience both the mileage and climbs asdifficult and tend to notice days on which both are high.The organizers want to convince cyclists to sign up andpay the trip costs so they can make money; a promise thatlong mileage days won’t have big climbs helps themrecruit cyclists. The staff have no vested interest eitherway.3.29a.b.The percentage of residents with a university degree appearsto be related to GDP. As the percentage with a universitydegree increases, so does GDP.c.It is possible that an educated populace has the skills tomake that country productive and profitable. Conversely, it121086420501520CanadaUnited KingdomGermanyJapanUnited StatesItalyFrance103025Percentage with university degreeGDPRelation Between Percentage with University Degreeand GDP (in trillions of $US)90001000200030004000500060007000800006050809070130120110100MilesClimbRelation Between Cycling Daily Mileage andCycling Daily Climb in Feetis possible that a productive and profitable country has themoney needed for the populace to be educated.3.30a.b.These data suggest that, although there are somefluctuations, a slight decrease in organ donation seems tohave taken place between 1994 and 2004.c.There are many possible answers to this question. As oneexample, you might be interested in how characteristics offamilies or types of deaths distinguish between agreeing toand declining to donate.3.31a.The independent variable is type of academic institution. Itis nominal; the levels are private national, public national,and liberal arts.b.The dependent variable is alumni donation rate. It is a scalevariable; the units are percentages, and the range of values isfrom 9 to 66.c.The defaults will differ, depending on which software isused.Here is one example.6050403020100NationalprivateNationalpublicLiberal artsAlumnidonationrate20101550 19941995199619971998199920002001200220032004YearOrgandonationrateOrgan Donation Rates per MIllion Deaths in Canada, 1994–2004

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APPENDIX CC-11d.The redesigns will differ, depending on which software isused. In this example, we have added a clear title, labeledthex-axis, omitted the key, and labeled they-axis (beingsure that it reads from left to right). We also toned downthe unnecessary color in the background and cut some ofthe extra numbers from they-axis. Finally, we removed theblack box from around the graph.e.These data suggest that a higher percentage of alumniof liberal arts colleges than of national private ornational public universities donate to their institutions.Moreover, a higher percentage of alumni of nationalprivate universities than of national public universitiesdonate.f.There are many possible answers to this question. Onemight want to identify characteristics of alumni whodonate, methods of soliciting donations that result in thebest outcomes, or characteristics of universities within agiven category (e.g., liberal arts) that have the highestrates.g.Pictures could be used instead of bars. For example,dollar signs might be used to represent the threequantities.h.If the dollar signs become wider as they get taller, as oftenhappens with pictorial graphs, the overall size would beproportionally larger than the increase in donation rate it ismeant to represent. A bar graph is not subject to thisproblem because graphmakers are not likely to make barswider as they get taller.3.32a.A Pareto chart is organized from the highest bar to thelowest bar, whereas a bar graph might be organized in anumber of different ways (e.g., alphabetical).b.The Pareto chart allows us to make comparisons moreeasily than does the bar graph. Moreover, we can veryeasily identify the countries with the highest and lowestGDP.6040200NationalprivateNationalpublicLiberal artsDonationrate %Type of universityAlumni Donation Rate3.33a.One independent variable is time frame; it has twolevels: 1945–1950 and 1996–1998. The otherindependent variable is type of graduate program; it alsohas two levels: clinical psychology and experimentalpsychology.b.The dependent variable is percentage of graduates who hada mentor while in graduate school.c.d.These data suggest that clinical psychology graduatestudents were more likely to have been mentored if theywere in school in the 1996–1998 time frame than if theywere in school during the 1945–1950 time frame. Theredoes not appear to be such a difference amongexperimental psychology students.e.This was not a true experiment. Students were notrandomly assigned to time period or type of graduateprogram.f.A time series plot would be inappropriate with so fewdata points. It would suggest that we could interpolatebetween these data points. It would suggest a continualincrease in the likelihood of being mentored amongclinical psychology students, as well as a stable trend,albeit at a high level, among experimental psychologystudents.g.The story based on two time points might be falselyinterpreted as a continual increase of mentoring rates forthe clinical psychology students and a plateau for theexperimental psychology students. The expanded data setsuggests that the rates of mentoring have fluctuated over theyears. Without the four time points, we might be seducedby interpolation into thinking that the two scores representthe end points of a linear trend. We cannot drawconclusions about time points for which we have no data—especially when we have only two points, but even whenwe have more points.3.34A pie chart would include the four percentages as slices, but itmight be difficult to make comparisons, particularly betweenpercentages similar in size (e.g., 9% and 5%). A bar graphallows for easier comparisons among categories than does apie chart.100806040200ClinicalExperimentalType of psychology graduate programPercentagePercentage of Mentoring by Time Frame andType of Psychology Graduate Program1945–19501996–1998

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C-12APPENDIX C3.35a.The details will differ, depending on the software used.Here is one example.b.The default options that students choose to override willdiffer. For the bar graph here, we (1) added a title,(2) labeled thex-axis, (3) labeled they-axis, (4) once wecreated the label on they-axis, we rotated it so that it readsfrom left to right, (5) eliminated the box around the wholegraph, and (6) eliminated the unnecessary key.3.36The examples will differ for each student. Correct answers willinclude the following types of variables.a.Frequency polygon: one scale variable; for example, on thex-axis, times for rats to complete a maze, and on they-axis,frequencies for each timeb.Line graph (line of best fit): two scale variables; forexample, on thex-axis, hours of maze-training for rats,and on they-axis, predicted times for rats to complete amazeCurrentRecentFormer0102030405060708090100Series 1100402080600PercentagePercentage Satisfied with Graduate AdvisorsAmong Current Students, Recent Graduates, andFormer Students Who Did Not Complete a PhDc.Bar graph (one independent variable): one nominal orordinal independent variable, such as gender of rat, on thex-axis, and one scale dependent variable, such as time tocomplete a maze, on they-axisd.Scatterplot: two scale variables; for example, on thex-axis,hours of maze-training for rats, and on they-axis, times forrats to complete a mazee.Time series plot: one time-related independent variable,such as year, on thex-axis, and one scale dependentvariable, such as mean GPA of incoming students, on they-axis,f.Pie chart: Trick question! Don’t use one; use a bar graphinstead.g.Bar graph (two independent variables): two nominal orordinal independent variables, such as gender of rat andreinforcement schedule for rat, on thex-axis, and a scaledependent variable, such as time to complete a maze, on they-axis3.37Each student’s advice will differ. The following are examples ofadvice.a.The shrinking doctor: Replace the pictures with bars.Space the 3 years out in relation to their actual values (inthe art shown, 1964 and 1975 are a good deal farther apartthan are 1975 and 1990). Make the main title moredescriptive.b.Workforce participation: Eliminate all the pictures. A fallingline in the art shown indicates anincreasein percentage;notice that 40% is at the top and 80% is at the bottom.Make they-axis go from highest to lowest, starting from0. Make the lines easier to compare by eliminating thethree-dimensional effect. Make it clear where the data pointfor each year falls by including a tick mark for each numberon thex-axis.3.38The articles and subsequent responses will be different for eachstudent.3.39a.The graph proposes that Type I regrets of action are initiallyintense but decline over the years, while Type II regrets ofinaction are initially mild but become more intense over theyears.b.There are two independent variables: type of regret (anominal variable) and age (a scale variable). There is onedependent variable: intensity of regrets (also a scalevariable).c.This is a graph of a theory. No data have been collected, sothere are no statistics of any kind.d.The story that this theoretical relation suggests is thatregrets over things a person has done are intense shortlyafter the actual behavior but decline over the years. Incontrast, regrets over things a person has not done butwishes they had are initially low in intensity but becomemore intense as the years go by.3.40a.This is a time plot. The researchers chose this type of graphbecause they wanted to show changes in the number ofpsychology degrees over time.b.This graph suggests a fairly large increase in bachelor’sdegrees over time, with smaller increases in master’s degreesand doctoral degrees.

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APPENDIX CC-13c.There are two independent variables. One is type of degree,with three levels: bachelor’s, master’s, and doctoral. It couldbe considered ordinal. The other independent variable isyear; it is a scale variable. The dependent variable is numberof psychology degrees; it is also a scale variable.d.There are several possible answers to this question. Forexample, they-axis starts at 0, there is a clear title, and alllabels read from left to right.e.There are several possible answers to this question. Forexample, the graph creator should have labeled thex-axis.They-axis is too “busy”; intervals of 10,000 would bebetter.f.There are several possible answers to this question. Forexample, we could track percentage out of all such degrees(e.g., percentage of psychology bachelor’s degrees conferredout ofallbachelor’s degrees conferred).g.There are several possible answers to this question. Forexample, we might examine what types of careerspsychology undergraduates pursue that do not require amaster’s or doctoral degree in psychology.3.41a.When first starting therapy, the client showed a decline, asmeasured by the Mental Health Index (MHI). After 8weeks of therapy, this trajectory reversed and there was aweek-to-week improvement in the client’s MHI.b.There are many possible answers. For example, the initialdecline in the client’s MHI may have been due todifficulties in adapting to therapy that were overcome as theclient and therapist worked together. Alternatively, it may bethat the client initially entered therapy due to difficult lifecircumstances that continued through the first weeks oftherapy but resolved after several weeks.c.Because the client is not beneath the failure boundary, andbecause the client experienced improvement over the lastfew weeks of therapy, it may be beneficial for the client tocontinue in therapy.3.42a.Density of traffic is represented by the thickness of thecolored lines across the roads. The flow of traffic isrepresented by the color of the lines.b.Answers to this question will vary, depending on the timeof day and the exact traffic conditions.c.This interactive graph allows anyone to see up-to-the-minute local, regional, and national traffic conditions.Traditional graphing techniques do not allow such up-to-date information on demand.3.43a.The independent variable is song type, with two levels:romantic song and nonromantic song.b.The dependent variable is dating behavior.c.This is a between-groups study because each participant isexposed to only one level or condition of the independentvariable.d.Dating behavior was operationalized by giving one’s phonenumber to an attractive person of the opposite sex. Thismay not be a valid measure of dating behavior, as we do notknow if the participant actually intended to go on a datewith the researcher. Giving one’s phone number might notnecessarily indicate an intention to date.e.We would use a bar graph because there is one nominalindependent variable and one scale dependent variable.f.The default graph will differ, depending on which softwareis used. Here is one example:g.3.44a.Sunday at midnightb.Saturday at 9A.M.c.Are people happiest on Saturdays? The independent variableis day of the week. The dependent variables are positiveattitude and negative attitude.d.Mood is operationalized as positive or negative attitudesexpressed in tweets sent via Twitter. This may not be a validmeasure of mood, as an expression of one’s attitude may notreflect someone’s actual mood states.e.This finding that people have the highest average negativemood on Sunday nights fits with the hypothesis that peopleare happiest on Saturdays. Because Sunday night is the nightbefore work or before school resumes for many people, wemight study whether the imminent start to the work andschool week is affecting people’s moods.C H A P T E R44.1The mean is the arithmetic average of a group of scores; it iscalculated by summing all the scores and dividing by the totalnumber of scores. The median is the middle score of all thescores when a group of scores is arranged in ascending order. Ifthere is no single middle score, the median is the mean of thetwo middle scores. The mode is the most common score of allthe scores in a group of scores.Romantic songNonromantic song0102030405060Percentage of Woman Who GavePhone Number Based on Song TypePercentage who gavephone numberSong typeRomantic songNonromantic song0102030405060%Gave Phone NumberBased on Song Type

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C-14APPENDIX C4.2The mean can be estimated by examining a visual display ofdata, such as a histogram, and finding the point in the data thatseems to create balance between the two sides of thedistribution. The mean can be calculated precisely usingarithmetic. In this case, you take all of the data points, sumtheir values, and then divide by the total number of scores.4.3The mean takes into account the actual numeric value of eachscore. The mean is the mathematic center of the data. It is thecenter balance point in the data, such that the sum of thedeviations (rather than the number of deviations) below themean equals the sum of deviations above the mean.4.4Unimodal distributions have one mode. Bimodal distributionshave two modes. Multimodal distributions have more than twomodes.4.5The mean might not be useful in a bimodal or multimodaldistribution because in a bimodal or multimodal distributionthe mathematical center of the distribution is not the numberthat describes what is typical or most representative of thatdistribution.4.6An outlier is an extreme score that is either very high or verylow in comparison with the rest of the scores in a sample.4.7The mean is affected by outliers because the numeric value ofthe outlier is used in the computation of the mean. Themedian typically is not affected by outliers because itscomputation is based on the data in the middle of thedistribution, and outliers lie at the extremes of the distribution.4.8The mode is typically used in one of three situations: (1) whenone score dominates the distribution, (2) to describe bimodalor multimodal distributions, and (3) when nominal data aresummarized.4.9The standard deviation isthe typical amount each score in adistribution varies from the mean of the distribution.4.10SD2comes from the wordsstandard deviation squared. Xrepresents the sample scores,Mthe sample mean, andNthenumber of scores in the sample. TheRindicates that valuesneed to be summed.4.11The standard deviation is a measure of variability in terms ofthe values of the measure used to assess the variable, whereasthe variance is squared values. Squared values simply don’tmake intuitive sense to us, so we take the square root of thevariance and report this value, the standard deviation.4.12a.1.m; SD22.The symbol for the mean should be capitalized, and thesecond symbol is for variance, not for standard deviation.3.M; SDb.1.l2.Greek letters are used for parameters; this is a statisticbecause it’s from a sample.3.Mc.1.XhighestXlowest2.The X’s should be italicized.3.XhighestXlowest4.13a.The mean is calculated as(15343246223634285228)/10327/1032.75MXNRThe median is found by arranging the scores innumeric order—15, 22, 28, 28, 32, 34, 34, 36, 46, 52—thendividing the number of scores, 10, by 2 and adding1⁄2to get5.5. The mean of the 5th and 6th score in our ordered listof scores is our median—(3234)/233—so 33 is themedian.The mode is the most common score. In these data,two scores appear twice, so we have two modes, 28and 34.b.Adding the value of 112 to the data from Exercise 4.15changes the calculation of the mean in the following way:(15343246223634285228112)/11439/1139.91The mean gets larger with this outlier.There are now 11 data points, so the median is the 6thvalue in the ordered list, which is 34.The modes are unchanged at 28 and 34.This outlier increases the mean by approximately 7values; it increases the median by 1; and it does not affectthe mode at all.c.The range is:XhighestXlowest521537The variance is:We start by calculating the mean, which is 32.7. Wethen calculate the deviation of each score from the meanand the square of that deviation.The standard deviation is:or4.14a.The mean is calculated as($44,751$52,000$41,500$38,862$51,380$61,774)/6$290,267/6$48,377.83The median is found by arranging the scores innumeric order: $38,862, $41,500, $44,751, $51,380,$52,000, $61,774. There are 6 scores, so the mean of the 3rdand 4th scores—($44,751$51,380)/2$96,131/2$48,065.50—is the median.5MXNR5255SDXMN()103.6110.182R5SDSD25255SDXMN()1036.110103.6122RXXM(XM)21517.7313.29341.31.69320.70.494613.3176.892210.7114.49363.310.89341.31.69284.722.095219.3372.49284.722.0952SDXMN()22R

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APPENDIX CC-15There is no mode among these data; all values occurjust once.b.Adding the value of $97,582 to the salary data in Exercise4.16 changes the calculation of the mean in the followingway:($44,751$52,000$41,500$38,862$51,380$61,774$97,582)/7$387,849/7$55,407There are now 7 data points, so the median is the 4thvalue in the ordered list, which is $51,380.There is still no meaningful mode.This outlier affects the mean twice as much as it doesthe median. There is no change to the mode.c.The rangeXhighestXlowest$61,774$38,862$22,912The variance isWe start by calculating the mean, which is $48,377.83. Wethen calculate the deviation of each score from the meanand the square of that deviation.The standard deviation isord.The range would change fromXhighestXlowest$61,774$38,862$22,912 toXhighestXlowest$97,582$38,862$58,7204.15a.The mean is calculated as[3.7(1.7)5.916.429.5 . . .1.7]/12244.2/1220.35°FThe median is found by arranging the temperatures innumeric order:3.7,1.7, 1.7, 5.9, 13.6, 16.4, 24, 29.5, 34.6, 38.5,42.1, 43.3There are 12 data points, so the mean of the 6th and 7thdata points gives us the median: (16.424)/220.2°F.b.The mean is calculated as[47(46)(38)(20) . . . 46]/12 163/12 13.58°F5MXNR5MXNRR5255SDXMN()$58,766,662.139$7,665.9425SDSD25255SDXMN()$352,599,972.8346$58,766,662.13922RXXM(XM)244,7513,626.8313,153,895.84952,0003,622.1713,120,115.50941,5006,877.8347,304,545.50938,8629,515.8390,551,020.58951,3803,002.179,013,024.70961,77413,396.17179,457,370.66952SDXMN()22R5MXNRThe median is found by arranging the temperatures innumeric order:47,46,46,38,20,20,5,2, 8, 9, 20, 24There are 12 data points, so the mean of the 6th and7th data points gives us the median: [20 5]/225/2 12.5°F.There are two modes: both46 and20 wererecorded twice.c.The mean is calculated as(173166180 . . .178)/122022/12168.5 mphThe median is found by arranging the wind gusts innumeric order:136, 142, 154, 161, 163, 164, 166, 173, 174, 178, 180, 231There are 12 data points, so the mean of the 6th and7th data points gives us the median: (164166)/2165 mph.There is no mode among these wind gusts.d.For the wind gust data, we could create 10-mph intervalsand calculate the mode as the interval that occurs mostoften. There are four recorded gusts in the 160–169 mphinterval, three in the 170–179 interval, and only one in theother intervals. So, the 160–169 mph interval could bepresented as the mode.e.The range is:XhighestXlowest43.3(3.7)47°FThe variance is:We start by calculating the mean, which is 20.35°F. We thencalculate the deviation of each score from the mean and thesquare of that deviation.The variance is:The standard deviation is:orf.The range isXhighestXlowest24(47)71°F5255SDXMN()275.97516.61°F2R5SDSD2R5255SDXMN()3311.69612275.97522XXM(XM)23.724.05578.4031.722.05486.2035.914.45208.80316.43.9515.60329.59.1583.72338.518.15329.42343.322.95526.70342.121.75473.06334.614.25203.063243.6513.32313.66.7545.5631.718.65347.82352SDXMN()22R5MXNR

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