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
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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
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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
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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
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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
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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
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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
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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
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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.
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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?
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