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Instructor’s Manual and Test Bank For Basic Statistical Analysis Ninth Edition Richard C. Sprinthall American International College i Introduction This Instructional Aid has been prepared to accompany Sprinthall’s “Basic Statistical Analysis”, ninth edition, and was written by Richard C. Sprinthall. The purpose of this manual is to aid the busy instructor. Large student enrollments, students with varying educational backgrounds, and a course that students often approach with trepidation….all combine to present the instructor in elementary statistics with a most difficult job. In fact, there is probably no other college course which is more demanding of the instructor's time than elementary statistics. Your students need time and attention, office-hour visits, extra sessions, and perhaps most of all, lots of problems to work out and immediate feedback. This all takes time, and in many cases, the instructor who has perhaps three (or more) other courses to prepare, simply doesn't have enough of that precious commodity ... time. Hopefully~ this manual will help ease some of the instructor's burden. Each chapter is divided into four parts. Part A involves suggestions for classroom activities. I certainly do not presume to tell any instructor how to teach this course, especially since most of you have had years of experience, not only in teaching statistics but also specifically teaching from previous editions of this text. There may some of you, however, who haven't taught the course before and these suggestions may help your preparations for class and hopefully provide a few ideas for keeping the class involved and interested. Too many students look forward to this course as though it were akin to a public flogging, but once you get them into it, once they recognize that these concepts have practical value in their lives, the course should then take on a higher interest level. There will always be some students who claim to have math blocks, and are even willing to proudly proclaim their innumeracy. These may be the same students who would never admit to even a smidgen of illiteracy, but somehow take perverse pride in their lack of numerical sophistication. "I may be innumerate, but I'm not illiterate". This is the same student who will fret over some technical grammatical point, but won't know how to find a percentage. One brief suggestion is to ask these students to clip out and bring to class newspaper or magazine articles that are devoted to proving some statistical or research point. Once they learn to become research critics, they may start carrying their calculators with a new sense of self esteem, and may even look forward to their visits to the computer lab. They may also discover that all newspaper accounts, even of the same story, aren't covered with the same attention to detail. In one analysis of how scientific stories are covered by the press, Evans et al. compared those in the New York Times and Philadelphia Inquirer on the one hand, with those in the National Enquirer or Star on the other (Evans et aI, 2009). The Times and Inquirer offered significantly more details than did the tabloids, but, alas, even the prestige newspapers omitted relevant information in the majority of their reports. We all know that feedback and constant practice are the essential ingredients for student success. The student who puts things off, and "lets it get away", may find it impossible to later catch up. Unlike a history course, where the student can spend an entire weekend ii doing catch-up reading, the student in statistics needs daily involvement. And daily student involvement translates into more work for the instructor, making up problems, doing the problems, and correcting the problems. The manual contains over 2000 test items, over 1200 multiple-choice items, over 400 true-false items and well over 400 problems requiring calculated solutions. Except for Chapters 1 and 11, which do not demand any numerical calculations, each chapter has a large number of problems where actual values are supplied. These may be used as homework problems, or as exam items. Further, both the multiple-choice and true-false items are odd-even balanced throughout each chapter. If the even-numbered items are used for the main exam, the odd-numbered items may be used for students needing make-up exams. The items are designed to be straight-forward and unambiguous. They are NOT intended to trick the student, but instead to provide an honest evaluation of the student's ability. There are no "curve balls" or "sliders"; instead, the questions all come "right down the middle", and although some may require a very careful reading, the student who is thoughtful and prepared should not find them overly difficult. One of the criticisms of the multiple-choice and true-false formats is that they reward rote memorization rather than true understanding. Although this can certainly at times happen, especially when the items are poorly designed, a carefully prepared multiple-choice test, such as the one presented in this manual, can assess a student’s ability to apply concepts to problem-solving situations. Rather than breaking up the unity of knowledge and isolating the pieces, as critics typically charge, a well- designed multiple-choice test demands that the student be able to understand concepts and bring facts together (Aiken, 1988). One fairly recent report showed that multiple-choice items may actually be more culturally fair than are essay questions. Verbally adept but uninformed students can sometimes bluff their way through essay exams, a tactic that is used less often by minorities (Science Agenda, 1990). Although some poorly prepared students will always insist on calling multiple-choice items, "multiple-guess tests", a careful analysis of student responses allows the instructor to illuminate a student's thought process in rather sharp detail (often to the amazement of the complaining student) . Finally, the manual also includes the answers to all the even numbered text problems (the oddnumbered answers are presented in the text itself). Aiken, L.R. (1988). Psychological testing and assessment (6th ed.). Boston: Allyn and Bacon. Evans,W. A., Krippendorf,M.,Yoon,J.H.,Posluszny,P. & Thomas, S. (2009). Science in the prestige and national tabloid presses. Social Science Quarterly, 71, 105-117. Science Agenda. (1990). New directions in educational testing. Science Agenda. (August-September), 14. iii CONTENTS TOPIC Introduction: PAGE i-xii CHAPTER 1: INTRODUCTION TO STATISTICS Suggestions for Class Activities Statistics and the Media Multiple Choice Items True-False 1 1 3 7 CHAPTER 2: PERCENTAGES, GRAPHS AND MEASURES OF CENTRAL TENDENCY Suggestions for Class Activities The Mean is still the Mean Assessing Central Tendency Your Students as Statistical Consultants Averaging Averages Evaluating Percentages The Mean and Adding a Constant The Mean and Multiplying by a Constant The Mean and Independent Measures Multiple Choice Items True-False Calculate the Values 8 CHAPTER 3: MEASURES OF VARIABILITY Suggestions for Class Activities Quartiles Anyone? How Normal is Normal? Skewing the Distribution Show Your Cash What About Constants? The SD and Multiplying by a Constant Multiple Choice Items True-False Calculate the Values 22 22 22 22 23 23 23 24 25 33 35 8 8 8 8 9 9 10 11 11 12 20 21 iv CHAPTER 4: THE NORMAL CURVE AND z SCORES Suggestions for Class Activities Means Z Scores and Skewness Small Samples and Z Scores 38 38 38 38 Remembering Z Scores Multiple Choice Items True-False Calculate the Values 39 40 49 51 CHAPTER 5: z SCORES, T SCORES AND OTHER NORMAL CURVE TRANSFORMATIONS Suggestions for Class Activities Curing the Algebra Phobe Translating Into T Scores Using Table 5.1 : Finding Stanines Multiple Choice Items True-False Calculate the Values 54 54 54 55 56 57 65 67 CHAPTER 6: PROBABILITY Suggestions for Class Activities The Lottery and Combinations Low Probabilities and the Impossible Multiple Choice Items True-False Calculate the Values 70 70 70 70 72 79 81 CHAPTER 7: STATISTICS AND PARAMETERS Suggestions for Class Activities Ask Me, and I Will Tell You Multiple Choice Items True-False Calculate the Values 84 84 84 85 93 95 v CHAPTER 8: PARAMETER ESTIMATES Suggestions for Class Activities Interval Estimates and Precision Interval Estimates, Hypothesis Testing and the Alpha Level 98 98 98 98 Multiple Choice Items True-False Calculate the Values 100 109 112 CHAPTER 9: THE FUNDAMENTALS OF RESEARCH METHODOLOGY Suggestions for Class Activities The Hawthorne Effect and John Henry Sensitization and Causal Inference A Research Example Reviewing the Literature The Student Researcher Multiple Choice Items True-False 116 116 116 117 117 118 121 123 145 CHAPTER 10:THE HYPOTHESIS OF DIFFERENCE Suggestions for Class Activities Student’s t as a Ratio Increasing the Variability Within Getting Creative with the Standard Deviation Multiple Choice Items True-False Calculate the Values 149 149 149 150 150 152 162 165 CHAPTER 11: THE HYPOTHESIS OF ASSOCIATION: CORRELATION Suggestions for Class Activities Correlation and the Cause-And-Effect Trap The Spearman rs (rho) Hard Data? Multiple Choice Items True-False Calculate the Values 171 171 171 172 172 174 185 187 vi CHAPTER 12: ANALYSIS OF VARIANCE Suggestions for Class Activities Comparing t and F F, t and the Direction of the Difference ANOVA and Independence 197 197 197 197 197 ANOVA and Post-Facto Research The Levene Test Multiple Choice Items True-False Calculate the Values 198 199 200 211 212 CHAPTER 13: NOMINAL DATA AND THE CHI SQUARE Suggestions for Class Activities Identifying Nominal Data Nominal and Interval: A Comparison Chi Square and the Hypothesis of Difference Chi Square and Correlation Multiple Choice Items True-False Calculate the Values 217 217 217 218 218 218 220 229 231 CHAPTER 14: REGRESSION ANALYSIS Suggestions for Class Activities Regression and Secular Trend Secular Trend and the Computer Trend Analysis and the Rear-View Mirror Multiple Choice Items True-False Calculate the Values 233 233 233 234 234 236 245 247 CHAPTER 15: REPEATED-MEASURES AND MATCHED-SUBJECTS DESIGNS (Interval Data) Suggestions for Class Activities 252 Degrees of Freedom and Marital Bliss 252 The Paired t and the Pearson r 252 Multiple Choice Items 255 True-False 265 Calculate the Values 267 vii CHAPTER 16: NONPARAMETRICS REVISITED - THE ORDINAL CASE Suggestions for Class Activities When Normality Fails Parametric Restrictions Non-Parametrics and their Parametric Counterparts 275 275 275 275 Multiple Choice Items True-False Calculate the Values 278 287 289 CHAPTER 17: TESTS AND MEASUREMENTS Suggestions for Class Activities Library Sources and the Internet Creating a Test Multiple Choice Items True-False Calculate the Values 294 294 294 295 296 301 302 SPECIAL UNIT 2 Multiple Choice Items 314 viii ANSWERS TO INSTRUCTOR TEST BANK (multiple choice, true-false and calculate the values). Chapter 1 317 Chapter 2 318 Chapter 3 319 Chapter 4 320 Chapter 5 322 Chapter 6 323 Chapter 7 324 Chapter 8 325 Chapter 9 327 Chapter 10 329 Chapter 11 332 Chapter 12 335 Chapter 13 337 Chapter 14 338 Chapter 15 339 Chapter 16 341 Chapter 17 343 Special Unit 2 345 ix COMPUTER PROBLEMS Chapter 2 346 Chapter 3 348 Chapter 8 349 Chapter 10 351 Chapter 11 356 Chapter 12 364 Chapter 13 370 Chapter 14 375 Chapter 15 381 Chapter 16 390 Chapter 17 393 x ANSWERS TO COMPUTER PROBLEMS Chapter 2 397 Chapter 3 398 Chapter 8 399 Chapter 10 400 Chapter 11 401 Chapter 12 402 Chapter 13 404 Chapter 14 405 Chapter 15 406 Chapter 16 407 Chapter 17 408 xi ANSWERS TO EVEN NUMBERED TEXT PROBLEMS Chapter 1 409 Chapter 2 411 Chapter 3 412 Chapter 4 413 Chapter 5 414 Chapter 6 415 Chapter 7 416 Chapter 8 417 Chapter 9 418 Chapter 10 420 Chapter 11 422 Chapter 12 424 Chapter 13 426 Chapter 14 427 Chapter 15 429 Chapter 16 430 Chapter 17 431 xii CHAPTER 1 INTRODUCTION TO STATISTICS A: SUGGESTIONS FOR CLASS ACTIVITIES Activity: Statistics and the Media Try to encourage your students to feel that statistics can help them understand current events as well as their own areas of specializations. Some of your students will devoutly believe that they are math dim-wits and will never understand these arcane statistical scribblings that you are putting up on the board. Impress your students with the fact that although the field of statistics has its detractors, and in many instances rightfully so, it is also an area that offers rich rewards to those with some degree of quantitative sophistication. At meetings, in journals and at conventions, students may find themselves bombarded with the latest findings of social research studies, and the results of these studies can often appear bewildering and at times even misleading if the student is totally unfamiliar with the general rules of statistical analysis and research design. Statistics and statistical research should be an aid to judgment, not a replacement for it. The kinds of topics covered in this first chapter lend themselves to lively discussions, usually based on current events reported in almost any newspaper or magazine. Ask the students to be on the alert for articles that attempt to use statistical persuasion to push for a point of view that may not really be backed up by the numbers. As one example tell them that a recent US administration took credit for creating wealth for American citizens during its eight years in office. The percentage of individuals earning over $50,000 a year, they said, had increased from 8% to 10%. But, by not factoring in inflation, that may not be (and isn't) any gain at all. $50,000 today does not have the purchasing power that $50,000 would have had eight years earlier. To stretch the point, tell them that 50 years ago very few Americans earned over $10,000, but does that mean we are all that much richer today? In the text, there is a discussion of a research study that attempted to prove that as women left the kitchen and entered the work force, terrible consequences ensued, such as dramatic increases in the prison population, small business failures and reported felonies. Students must be constantly reminded that just because various events might occur together they are not necessarily linked and one should not be led into the trap of a cause-and-effect explanation. On June 20, 1993 the New York Times ran an ad that advised its readers that virtually all of society's problems can be blamed on TV. The ad referred to a study by B.S. Centerwall at the Univ. of Washington that showed that in the 30 year period following the introduction of TV in the United 1 States and Canada, white homicide rates have increased by 93%, and in Canada by 92%, whereas in South Africa, which had no TV until the mid 1970's, the white homicide rate decreased by 7%. Perhaps your students might comment on this artful juxtaposition of statistics, and you might point out that a whole host of things may have changed during that time span. And as for South Africa, maybe whites were not being killed at the same rate, but that in itself doesn't prove that there were fewer overall numbers of homicides, with or without TV. Finally, it might be mentioned that some of history's most notorious murderers, Hitler, Stalin, Al Capone and Jack the Ripper, never saw one second of TV. 2 B. Multiple Choice Items 1-1. The inherent fallacy in the argument that capital punishment is not a deterrent (since pockets were being picked even at the public hanging of a pickpocket) is that a. pickpockets were never hanged b. you can't compare groups of unequal sizes c. there was no comparison or control group d. you can't make comparisons with only nominal data 1-2. Whenever two events occur simultaneously, such as increasing numbers of joint bank accounts and increasing numbers of felonies, one must be careful not to a. assume that one event is the cause of the other b. assume that any correlation exists between them c. assume that the data have been reported honestly d. all of these 1-3. The founding of probability theory is popularly credited to a. the Chevalier de Mere b. Sir Francis Galton c. Karl Pearson d. Blaise Pascal 1-4. William Sealy Gossett, the statistician at the Guinness Brewing Company, published his works using the pen name a. Professor b. Blaise Pascal c. Cicero d. Student 1-5. Statistics as a general field is divided into two sub-areas. They are a. predictive and inferential b. descriptive and inferential c. nominal and ordinal d. none of these 1-6. When we say that the actual average height of all adult females in the U.S. is known to be 5' 6", we are using a. predictive statistics b. probability statistics c. inferential statistics d. descriptive statistics 3 1-7. Techniques which are used for describing small or large amounts of data in abbreviated form, are called a. b. c. d. descriptive statistics inferential statistics predictive statistics probability estimates 1-8. The difference between inferential and predictive statistics is that a. inferential extrapolates, whereas predictive does not b. predictive extrapolates, whereas inferential does not c. inferential assumes that a sample has been measured, whereas predictive makes no such assumption d. there is no difference between them 1-9. When a researcher attempts to estimate the characteristics of an entire population on the basis of sample measures, the techniques employed are called a. inferential statistics b. descriptive statistics c. skewed statistics d. b and c, but not a 1-10. The researcher using inferential statistics always makes predictions that are based on having measured a. a population b. a sample c. the entire group being predicted d. in inferential statistics predictions are never made 1-11. The goal of the researcher using inferential statistics is to a. make better-than-chance predictions b. predict the characteristics of the entire group, based on measures taken on a smaller group c. describe the difference between the highest and lowest score d. a and b, but not c 1-12. An example of a variable would be a. height b. weight c. the number of inches in a foot d. a and b, but not c 4 1-13. Anything that can be measured and observed to vary is called a. a constant b. a variable c. an integer d. all of these 1-14.The statement that one can prove anything with statistics is only true when a. the data have been faked b. the reader is naive regarding statistical procedures c. the statement is always true d. the statement is never true 1-15. The first attempt to employ statistical techniques for estimating population parameters was designed in order to a. estimate beer-drinking tastes in Dublin, Ireland b. calculate the odds for casino gambling in France c. calculate the probability of certain exponential functions d. all of these 1-16. The first attempt to employ probability theory in a practical setting was developed in order to a. predict the results of presidential elections b. predict economic forces in the market place c. predict how consumers will evaluate new products d. make winning bets in the gambling casinos 1-17. One author suggests that the best way to compare the safety of autos versus planes is on the basis of a. the total number of deaths per year for each mode of travel b. the total number of individuals who drive versus fly. c. the per-hour death rate for the two modes of travel d. none of these, since travel rates are not comparable. 1-18. In comparing the safety of allowing children to visit friends whose parents own a gun versus those parents who have a swimming pool, it was found that a. there was no safety difference between the two b. children were safer at the homes with swimming pools c. children were safer at the homes with guns d. more parents had guns than swimming pools 5 1-19. Since poll results are based on a sample size which is less than the population size, generalizing from the sample to the population demands a. data graphing b. inferential statistics c. descriptive statistics only d. leap of faith 1-20. The commercial that specifies that a certain brand of aspirin should be used, since no other brand prevents more heart attacks proves that a. the brand mentioned prevents more heart attacks than does its competition b. the brand mentioned has been proven to prevent heart attacks c. the brand mentioned has only been shown effective when combined with exercise d. none of the above 6 C. True or False: For the following, indicate T (true) of F (false) 1-21. Descriptive statistics are designed to infer population values. 1-22. Descriptive statistics provide symbolic techniques for describing both large and small groups of data. 1-23. Inferential statistics involve the measuring of a sample and then using this value for predicting to the population. 1-24. Inferential statistical techniques and predictive statistical techniques are identical. 1-25. Descriptive statistics are especially important in using polling data to infer population characteristics. 7 CHAPTER 2 Percentages, Graphs and Measures of Central Tendency A: SUGGESTIONS FOR CLASS ACTIVITIES Activity: The Mean is still the Mean Although in the text the decision was made to use the symbol M for the arithmetic mean, point out to the class that this is not the only way the mean can be expressed. Although most of the Education and all of the Psychology journals now being published, use M for the arithmetic mean, there continues to be a small number of statistics texts that are still using X with a bar across the top. Students may at first be confused by this inconsistency, but once it is pointed out that both symbols mean exactly the same thing, the mean is the mean is the mean, the level of possible frustration should be reduced. In fact, you may wish to accept either symbol. Activity: Assessing Central Tendency: What do both the Median and Mean mean? Perhaps the most glaring trap awaiting students who misunderstand central tendency is the confusion that arises between the use of the mean and the median. Both measures, to be sure, provide information regarding how the average or typical subject performed, but in certain situations the use of one of these measures rather than the other can create an extremely inaccurate portrayal of centrality. . Activity: Your Students as Statistical Consultants Ask your students to assume that have been selected as statistical consultants and have been given the following scores on a standardized test of reading ability test (where a score of 100 indicated normal progress): X 110 109 109 108 107 107 107 15 X=772 For this distribution, then, X = 772, and the mean of X = 772/8 = 96.50. On the basis of this mean value of only 96.50 it would seem that on average the group was not performing up to the standards of normal progress, even though 8 every single student in the group, except for one, was scoring well above the average. In this case, as with all skewed distributions, the median, which for this distribution is 107.50, is a far more accurate indicator of true centrality than was the mean. Point out that when the distribution is skewed to the left, as in the above scores, the mean is going to severely underestimate the true centrality. Have the students graph the above distribution to again reinforce what the shape of a skewed distribution looks like. If equal intervals are chosen for the base line (abscissa), the graph will have to be extremely wide to fit all the values in. Also, point out that the median remains at 107.50 whether the low score had been 15 or 105. However if the low score were changed to 105 the mean would then jump to 107.75. Changing that one score caused the mean to gyrate, but the median remained rock steady at 107.5. Activity: Averaging Averages? Sometimes students will intuitively assume that to get the mean of two sets of scores, all they have to do is average the two means. This of course is only true if both sets of scores have equal numbers of cases. But show them that with unequal numbers of cases, averaging the means can be a big mistake. For example, the following distribution 16,12,12,11,10,9,9,7,4 adds up to 90, with a mean of 10. A second distribution, 10,8,8,8,8,6 adds up to 48, with a mean of 8. The mean of the two distributions combined is 9.20, not the average of the two means which would have been 9. You can, however, teach students to do this correctly without going back and adding all the scores. Show them that since the mean, M, is = to X/N, then X = the M times N, or (M)(N). X for the first distribution is then equal to (M)(N) or (10)(9) = 90. Similarly for the second distribution, X = (M)(N) = (8)(6) = 48. They will quickly see that the mean of both distributions combined can easily be found by adding the two Xs (48+90) = 138 and dividing by the total N of 15, to get 138/15 = 9.20. If they think this is complicated and would rather just put all the scores together and add them up, explain that with a large data base, the technique you're showing them is far more efficient. Activity: Evaluating Percentages The same concerns also tend to show up when evaluating percentages. Too often students want to average percentages, even when the totals in the various percentage categories are not the same. Even faculty members have been known to have difficulty accepting the fact that means and percentages cannot always simply be averaged. On a Master's Comprehensive exam at a small eastern college, the passing grade on the objective section of the test was determined to be 80% correct. This section was composed of 300 multiple choice items, covering seven different content areas, but the seven content areas were not all composed of the same number of items. For example, the exam could have had 9 100 items devoted to Learning, 100 to Systems and Theories, and then 20 items each in Cognitive Psychology, Psychological Assessment, WISC-Assessment, Statistical Analysis and, finally, Learning Disabilities. A student could then have received scores of 95% in Learning, 85% in Systems, 60% in Cognitive, 60% in Assessment, 60% in WISC, 60% in Statistics and 60% in LD. The faculty group challenged the fact that the student's overall score could have resulted in a passing grade of 80%. The scoring breakdown was as follows, 95 out of a hundred in Learning, 85 out of a hundred in Systems, and then 12 out of 20 for the other five sections. This resulted in a total of 240 correct responses out of 300 items, or 80% correct. Thus, the student could have passed the exam, even though failing in 5 of the 7 sections. Or cite the example of the baseball player who hit .300 in day games and only .200 in night games. This player wondered why he was being sent down to the minors when his day-night average was a seemingly adequate .250. The problem was that the team had only played 5 day games but had already played over a 100 night games, and for all 105 games his average was a mere .215. By the way, this situation was seriously argued on a sports, call-in radio show. Activity: The Mean and Adding a Constant Explain that the effect on the mean of adding a constant to every value is to simply change the mean by the amount of that constant. Thus, the new mean = the old mean + the constant. In the following set of scores: X 14 12 3 9 8 10 5 2 63 Mean = 63/8 = 7.875 Now we will add the constant 10 to each of the previous values X+10 24 22 13 19 18 20 15 12 143 Mean = 143/8 = 17.875 (or 7.875 plus the constant 10). 10 Activity: The Mean and Multiplying by a Constant Show your students that multiplying by a constant has the effect of changing the mean by a function of that constant, such that the new mean equals the old mean times the constant. Using that first distribution shown above, each value will be multiplied by the constant 10 (X)(10) 140 120 30 90 80 100 50 20 630 Mean = 630/8 = 78.750 (or 7.875 times the constant 10). Activity: The Mean and Independent Measures Let the students see what happens to the mean when two independent measures are summed. When the mean is being found for the sum of two measures, for example if you have two independent measures on each subject, and these measures are added, then X1 + X2 = X1+X2 15 + 11 = 26 14 + 10 = 24 12 + 9 = 21 11 + 5 = 16 10 + 5 = 15 9 + 4 = 13 7 + 3 = 10 2 + 1 = 3 80 48 128 m1 = (80/8 = 10.00) + m2= (48/8 = 6.00) = M for 128/8 = 16.00 Thus, the mean of the sums (16.00) is equal to the sum of the two means (10.00+6.00). 11 B. Multiple Choice Items 2-1. When scores are arranged in order of magnitude, the researcher has formed a a. histogram b. measure of centrality c. measure of dispersion d. distribution 2-2. Traditionally, the researcher indicates frequency of occurrence on the graph's a. ordinate b. abscissa c. line of ascent d. horizontal axis 2-3. When single points are used to designate the frequency of each score, the points being connected by a series of straight lines, this is called a a. frequency polygon b. frequency rectangle c. scatter plot d. histogram 2-4. The mean, median, and mode are all measures of a. dispersion b. variability c. central tendency d. all of these 2-5. When a graph is constructed using a series of rectangles indicating the frequency of occurrence for each score, it is called a a. frequency polygon b. frequency rectangle c. scatter plot d. histogram 2-6. The measurement which occurs most often in a distribution is called the a. median b. percentile c. mean d. mode 12 2-7. When a distribution is skewed, the researcher who is interested in central tendency should use the a. mean b. median c. mode d. all of these are appropriate 2-8. When a distribution shows a large majority of very low scores and a few very high scores, the distribution is said to be a. skewed to the right b. skewed to the left c. skewed to the middle d. bimodal 2-9. The influence of a few extreme scores in one direction is most pronounced on the value of the a. mean b. median c. mode d. percentile 2-10. Using the mean to indicate centrality on a distribution of income scores usually results in a. a false image of poverty b. an accurate portrayal of income c. a false image of prosperity d. income scores never lend themselves to centrality 2-11. When each score is listed in order of magnitude, together with the number of individuals receiving each score, the researcher has set up a. a unimodal distribution b. a bimodal distribution c. a skewed distribution d. a frequency distribution 2-12. The abscissa is a. the horizontal axis b. the vertical axis c. the connected points on a polygon d. a measure of central tendency 13 2-13. On a frequency distribution, raw scores are plotted on the a. b. c. d. abscissa ordinate vertical axis all of these, depending on the size of the group being measured 2-14. When graphing data, it is traditional to make the length of the ordinate equal to a. the length of the abscissa b. twice the length of the abscissa c. three-quarters of the length of the abscissa d. one-half of the length of the abscissa 2-15. With a frequency polygon, scores are always presented on a. the X axis b. the Y axis c. the Z axis d. the frequency polygon may never be used to represent scores 2-16. The more separate scores there are in a given distribution, the higher will be the value of the a. the mean b. the median c. the mode d. none of these 2-17. The ordinate is identical to the a. X axis b. Y axis c. mean d. none of these 2-18. The so-called "wow" graph is always possible whenever a. scores are presented on the X axis b. the abscissa does not begin with zero c. the base of the ordinate is not set at zero d. two distributions are being presented simultaneously 2-19. Perhaps the most serious flaw in graphing data is due to a. not placing frequencies on the abscissa b. not placing raw scores on the ordinate c. not placing the ordinate on the X axis d. not setting the base of the ordinate at zero 14 2-20. The following are all measures of central tendency, except a. the mean b. the median c. the range d. the mode 2-21. The arithmetic average defines the a. mean b. median c. sigma d. mode 2-22. The point above which half the scores fall and below which half the scores fall, defines the a. mean b. median c. sigma d. mode 2-23. The most frequently occurring score in the distribution defines the a. mean b. median c. sigma d. mode 2-24. The mean is not overly affected by extreme scores, unless a. the extreme scores are all in one direction b. the extreme scores are in both directions c. the number of extreme scores is fewer than 5 d. all of these 2-25. The fact that the mean IQ of college seniors is higher than that of freshmen is probably due to a. the fact that going to college increases the IQ b. the fact that there is a big IQ gain between the junior and senior years c. an incorrect interpretation of the data d. the fact that the lower IQ freshmen tend to drop out of college and, therefore, never become seniors 2-26. Adding just one or two extreme scores to the high end of a distribution, has a great effect on a. the median, but not the mode b. the mode, but not the mean c. the mean, but not the median d. none of these 15 2-27. Adding just one or two extreme scores to the low end of a distribution, has a great effect on a. b. c. d. the median but not the mode the mode, but not the median the mean, but not the median none of these 2-28. When the majority of scores are at the high end of the distribution, but there are a few extremely low scores, the distribution is a. bimodal b. multimodal c. skewed left d. skewed right 2-29. When the mean lies to the right of the median, the distribution is probably a. bimodal b. multimodal c. skewed left d. skewed right 2-30. When the median lies to the right of the mean, the distribution is probably a. bimodal b. multimodal c. skewed left d. skewed right 2-31. When a distribution is skewed to the right, a. the mode will be to the left of the median b. the mode will be to the right of the median c. the mode will be to the right of the mean d. the mode will always be identical to the mean 2-32 Percentages are based on a standardized denominator of a. 100 b. 10 c. 50 d. 0 2-33 In order to read a percentage a. only the numerator of the percentage needs to be shown b. the percentage is always shown in fraction form c. the percentages shown are always in the form of inferential statistics d. to establish a percentage for a specific event, the total number of events need not be known 16 2-34 When comparing percentage rate increases with decreases, the same absolute difference yields a. the same percentage difference b. the percentage increase calculates out as larger than the decrease c. the percentage decrease calculates out as larger than the increase d. comparing percentage increases with decreases cannot be done 2-35 The FBI’s Uniform Crime Reports provide per capita data based on a rate per a. 100,000 b. 50,000 c 25,000 d. one million 2-36 Bar charts are used instead of histograms when the data are Continuous b. Non-continuous c. In the form of values that may fall at any point along an unseparated scale of points d. None of these since bar charts and histograms are synonymous. Questions 37 through 42 are based on the following: In a certain community, the median per-family annual income is $80,000. The Mean per-family income is $100,000, whereas the mode is $71,000. 2-37. the distribution of income scores is A. skewed right B. skewed left C. skewed to the middle D. not skewed 2-38. the most appropriate measure of central tendency in this distribution Would yield a value of A. $80,000 B. $100,000 C. $71,000 D. none of these values could yield a measure of central tendency 2-39. if a new family were to move into the community with an annual income of $295,000, this would most affect A. the mean B. the median C. the mode D. all of these 17 2-40. the annual income achieved by most of the families is A. $71,000 B. $80,000 C. $100,000 D. half way between the mean and the mode 2-41. The annual income which is surpassed by 50% of the families is a. $80,000 b. $71,000 c. $100,000 d. cannot tell from these data 2-42. The annual income which is surpassed by 90% of the families is a. $100,000 b. $71,000 c. $80,000 d. cannot tell from these data 2-43. Whenever a distribution is skewed left, the measure yielding the highest numerical value is always the a. mean b. median c. mode d. percentile 2-44. When a skewed distribution tails off to the right, the distribution is a. skewed right b. skewed left c. skewed to the center d. not skewed at all 2-45. In a histogram, the mode is always located a. under the shortest bar b. under the tallest bar c. under the last bar to the right d. under the last bar to the left 2-46. A bimodal distribution often indicates a. that there will be two means b. that there will be two medians c. that the mean, median and mode have the same value d. that two separate sub-groups may have probably been measured 18 2-47. The most appropriate measure of central tendency in a bimodal distribution is (are) the a. b. c. d. mean median modes ordinate 2-48. When a distribution has two separate and distinct medians, then a. it is skewed right b. it is skewed left c. it is probably bimodal d. a distribution can never have more than one median 2-49. With a fairly balanced distribution of (neither skewed nor bimodal), the most appropriate measure of central tendency is the a. mean b. median c. mode d. none of these 19 C. True or False: For the following, indicate T (True) or F (False) 2-50. A skewed right distribution has the mean lower than the mode. 2-51. The median is always exactly half-way numerically between the highest and lowest scores. 2-52. The most appropriate measure of central tendency in a skewed right distribution is the median. 2-53. A positively skewed distribution is identical to a skewed right distribution. 2-54. Other things being equal, the mean is the most stable measure when the data form is skewed.. 2-55. With a skewed left distribution, the median is always to the right of the mean. 2-56. With a skewed left distribution, the mode is never to the left of the mean. 2-57. All three measures of central tendency can be calculated when the data are in interval form. 2-58. On a frequency distribution curve, frequency of occurrence is always plotted on the abscissa. 2-59. One should expect a distribution of personal income measures to be skewed to the right. 2-60. When the median is being calculated, it makes no difference whether one starts counting from the bottom or the top of the distribution. 2-61. If a positively skewed and negatively skewed distribution were combined, the resulting distribution would probably be bimodal. 20 D. For the following questions, calculate the values. 2-62. For the following set of scores, calculate the mean, median and mode: 11, 2, 3, 3, 7, 6. 2-63. For the following set of scores, calculate the mean, median and mode: 20, 8, 18, 10, 15, 10, 13, 11. 2-64. For the following set of scores, calculate the mean, median and mode: 3, 4, 7, 7, 5, 9. 2-65. For the following set of scores, calculate the mean, median and mode: 5, 7, 3, 9, 4, 5, 10, 9. 2-66. For the following set of scores, calculate the mean, median and mode: 8, 8, 6, 7, 5, 9. 2-67. For the following set of scores, calculate the mean, median and mode: 12, 1, 9, 7, 2, 4. 2-68. For the following set of scores, calculate the mean, median and mode: 10, 12, 9, 10, 10. Questions 69 through 75 are based on the following: Thirty-four members of a certain sorority were selected, and asked to indicate how many hours each had spent reading (for pleasure, not school work) during the previous week. The data are as follows: 50, 4, 10, 5, 5, 6, 7, 3, 5, 4, 4, 5, 6, 6, 7, 5, 8, 1, 8, 7, 5, 6, 10, 6, 8, 7, 7, 6, 5, 5, 4, 3, 4, 5. 2-69. Find the mean. 2-70. Find the median. 2-71. Find the mode. 2-72. Which measure of central tendency yielded the highest numerical value? 2-73. Which measure of central tendency yielded the lowest numerical value? 2-74. If the distribution is skewed, in what direction is the skew?. 2-75. What would the mean and median have been if the highest score had been a 12 instead of a 50? 21