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Journal of Education for Business

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Evaluating the Use of Random Distribution Theory
to Introduce Statistical Inference Concepts to
Business Students
Karen H. Larwin & David A. Larwin
To cite this article: Karen H. Larwin & David A. Larwin (2011) Evaluating the Use of Random
Distribution Theory to Introduce Statistical Inference Concepts to Business Students, Journal of
Education for Business, 86:1, 1-9, DOI: 10.1080/08832321003604920
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Date: 11 January 2016, At: 22:09

JOURNAL OF EDUCATION FOR BUSINESS, 86: 1–9, 2011
C Taylor & Francis Group, LLC
Copyright

ISSN: 0883–2323
DOI: 10.1080/08832321003604920

Evaluating the Use of Random Distribution Theory
to Introduce Statistical Inference Concepts to
Business Students
Karen H. Larwin
Downloaded by [Universitas Maritim Raja Ali Haji] at 22:09 11 January 2016

Youngstown State University, Youngstown, Ohio, USA

David A. Larwin
Kent State University at Salem, Salem, Ohio, USA

Bootstrapping methods and random distribution methods are increasingly recommended as
better approaches for teaching students about statistical inference in introductory-level statistics courses. The authors examined the effect of teaching undergraduate business statistics
students using random distribution and bootstrapping simulations. It is the first such empirical
demonstration employing an experimental research design. Results indicate that students in the
experimental group—where random distribution and bootstrapping simulations were used to
reinforce learning—demonstrated significantly greater gains in learning as indicated by both
gain scores on the Assessment of Statistical Inference and Reasoning Ability and final course
grade point averages, relative to students in the control group.
Keywords: bootstrapping, random distribution, statistical inference

INTRODUCTION
Discovering how students learn most effectively is one of
the major goals of research in education. Over the last 30
years, many researchers and educators have called for reform in the area of statistics education in an effort to more
successfully reach the growing population of students across
an expansive variety of disciplines, who are required to complete coursework in statistics (e.g., Garfield, Hogg, Schau,
& Whittinghill, 2000; Higgins, 1999). Many of these students have very little interest in learning mathematics, and

even less interest in learning statistics (Bradstreet, 1996;
Gordon, 1995; Hollis, 1995). In light of this, reform efforts
have proposed that statistics education should abandon the
“information transfer model in favor of a constructivist approach to learning” (Moore, 1997, p. 124) in an effort to help
students develop an understanding of statistical concepts, beyond the use of mathematical formulas.

Correspondence should be addressed to Karen H. Larwin, Youngstown
State University, Department of Foundations, Research, Technology and
Leadership, One University Plaza, Youngstown, OH 44555–0001, USA.
E-mail: drklarwin@yahoo.com

The constructivist approach to learning is based on the
premise that learning is the result of mental constructions in
which students are able to incorporate new information by
building on knowledge they already have acquired (Caine
& Caine, 1991). Moore (1997) maintained that statistics is
not a subcategory of mathematics, but rather it is a science,
much like physics is a science. Statistics, like physics, can
depend heavily on mathematical computations; however, unlike mathematics, statistics is the science of inference with
different modes of thinking and concepts distinct from mathematics. Moore posited that successful statistics education

for the nonmath majors should include a balance of content,
pedagogy, and technology.
Statistics is an important core course in most undergraduate business programs in the United States (Sirias, 2002).
Although statistics is a core course in the sequence of business classes, many business students do not understand its
relevance to their education or future job prospects (Zanakis
& Valenzi, 1997). Many business students consider required
statistics coursework to be “irrelevant to their discipline, difficult, unattractive, and boring” (Selvanathan & Selvanathan,
1998, p. 1352). As a result, statistics professors are challenged to present the required course content in such a way
that students are motivated to engage material they initially

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K. H. LARWIN AND D. A. LARWIN

believe is uninteresting and irrelevant (Conners, McCown,
& Ewoldsen, 1998). In light of the significance of statistics
and statistical reasoning in the field of business, pedagogical approaches that engage and entice this otherwise notso-interested audience of future business graduates must be
examined and researched.

Statistics is not a spectator sport, and learning about statistics should not be either. Over the last 10 years, research in
statistics education has revealed that cooperative, activitybased, and technology-assisted learning are the primary pedagogical approaches that have the potential to enhance students’ understanding, and, in effect, enhance student’s ability
to effectively apply statistical concepts (Fernandez & Liu,
1999; Garfield, 2003; Garfield et al., 2002; Ware & Chastain,
1991; Yesilcay, 2000). Research indicates that a majority
(over 66%) of the postsecondary level statistics professors
surveyed in 2002 reported using constructivist approaches
(Garfield et al.). However, although some studies examining
the use of constructivist- or activity-based approaches to the
teaching of statistics have shown some promise in improving
student perceptions (e.g., Fernandez & Liu; Ware & Chastin;
Yesilcay), more recent research (e.g., Onwuegbuzie, 2004)
indicates that approximately 80% of students surveyed continue to experience negative feelings and bad attitudes about
their statistics education.
Recently, Rossman, Chance, Cobb, and Holcomb (2008)
proposed that statistics education should move from the
Ptolemaic curriculum, based on estimation procedures supported by normal distribution theory, to the now technologically enabled random distribution theory approach. Traditionally delivered statistics education is built around the
concept of a normal distribution as approximating a sample
distribution. The results that are computed are, at best, approximations. But, with the availability and power of computers today, Cobb (2007) maintained that statistics education
should now evolve to the presentation of random distribution

theory concepts, based on permutation tests, as the central
paradigm for statistical inference (p. 12). He maintained that
this approach is “simple and easier to grasp” (Cobb, p. 12)
for the novice, and can encourage students to more easily
embrace the logic of inference. Cobb, like Moore (1997,
2001), suggested that understanding the science of inference
is fundamental to an authentic understanding of statistics and
statistical applications.
Specifically, the random distribution theory approach allows for the rerandomizing of all possible combinations of
outcomes to see what is typical and what is not. According to Moore and McCabe (2005), resampling procedures
represent one of the single most important developments in
statistics education in a generation, without changing the
fundamental reasoning of statistical inference. Resampling
provides the teacher and learner with a visual presentation of
random samples from the population that is not hindered by
the need for normally distributed or large samples. Resampling helps to develop students’ understanding by providing

a medium through with students can carry out repetitions,
while controlling for the number of repetitions as well as the
sample size (del Mas, Garfield, & Chance, 1999). Once these

repetitions are completed, students can describe and explain
the behavior that they have observed with their data. Moore
and McCabe maintained that these procedures are intuitively
more appealing because they “appeal directly to the basis of
all inference: the sampling distribution demonstrates for the
student what would happen if we took many repeated samples
under the same conditions” (p. 2). Researchers have found
that with simulations abstract concepts such as sampling distributions, confidence intervals, and conclusions regarding
statistical significance become more conceptually clear to
students (Rossman & Chance, 2006).
Rossman et al. (2008) developed a number of applet-type
simulations that can facilitate the use of the random distribution theory approach in the classroom. Each of the learning modules developed by Rossman et al. guide students
through the ideas of randomization, repeating the random
selection process through resampling, and making decisions
as to whether or not the null hypothesis is plausible or should
be rejected. This approach to introducing statistical inference
provides students with a better chance of developing an understanding of how to interpret the results of a study based on
null hypothesis statistical testing; more specifically, to help
them understand what p values indicate. Students use these
applet-type simulations to construct for themselves an understanding of the connection between a randomly designed

experiment, and the conclusions that result from the statistical analysis.
An example of one of the applets used in this investigation is presented in Figure 1 (http://www.rossmanchance.
com/applets/Dolphins/Dolphins.html). In this simulation activity, the students are asked to consider a study conducted by
Antonioli and Reveley (2005). The study examines the effectiveness of dolphin-mediated water therapy, relative to traditional group therapy, in the treatment of mild to moderate depression. Students are presented with the results of the study
and asked to explore whether it is possible that these findings
indicate that dolphin-mediated water therapy resulted in significantly greater number of patients who showed substantial
improvement, or whether the findings were simply the result
of chance variation.
As can be seen in Figure 1, the observed results of the
study are presented to the student. The simulation provides
the student with the ability to repeatedly resample the 30 individuals into the experimental and control group conditions,
in an effort to see if resampling produces results as extreme
as 10 improved patients in the experimental group, as was
the case in the original study. The graph on the right half
of Figure 1 depicts the results of one such resampling (n =
1000) and illustrates that the result of 10 patients showing
improvement is clearly a rare outcome.
In the present study we sought to examine the impact of
random distribution theory-based applications proposed by

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RANDOM DISTRIBUTION THEORY

FIGURE 1

3

Dolphin Applet example.

Cobb (2007) and developed by Rossman et al. (2008) relative
to the impact of assignments based on normal distribution
theory, on students’ understanding of statistical reasoning
and statistical inference. Specifically, half of the students in
an activity-based, cooperative learning facilitated undergraduate business statistics class were assigned to complete eight
two-part sections of Rossman et al.’s learning modules. Their
peers were asked to complete an assignment of equal difficulty level and time-commitment that was based on more
traditional approaches. The goal of this investigation was to
see if the introduction of the random distribution simulations
did in fact impact student understanding of statistics and

statistical inference, specifically in a statistics section that
was constructed as an activity-based learning community. At
present, there are no published studies in the available literature in which the researchers employed an experimental
design examining whether random distribution theory can
facilitate students’ understanding of statistics and statistical
inference. The present study is the first such attempt.

METHOD
Participants
Participants included second- and third-year undergraduate
business students that ranged from 18 to 52 years of age
(M = 20.82, SD = 5.08), including 33 men (59.6%) and
21 women (40.4%). These university students were enrolled
in a Monday-Wednesday-Friday morning business statistics
course section. Students in this course were expected to develop the statistical tools used in business decision making,
including but not limited to determination and interpretation
of measures of central tendency, variance, probability, regression and correlation analysis, hypothesis testing, frequency
and probability distributions, and sampling issues. Students

were also introduced to graphical, tabular, and mathematical

depictions of statistical information.
Instrumentation
The Assessment of Statistical Inference and Reasoning Ability (ASIRA) was used to assess the students’ inferential and
reasoning skills. Specifically, the ASIRA was constructed
of 20 questions intended to measure both statistical reasoning skills and understanding of statistical inference. The first
13 questions were adapted from the Statistical Reasoning
Assessment (SRA) developed by Joan Garfield (2003) and
further adapted for computer-administrated assessment. The
last seven questions were adapted from an assessment developed by Rossman et al. (2008), and were intended to
measure a student’s understanding of statistical inference.
This 20-question assessment took students approximately
25 min to complete. The same assessment was used for both
the pre- and posttest measures. The Cronbach’s alpha, a test
of internal consistency, for the pre and posttest measures was
0.79 and 0.84, respectively. A copy of the ASIRA inventory
is found in Appendix A.
The student’s final course grade (based on 400 possible
points) also was used to assess learning. The final course
grade included 4 exam scores, 10 quiz scores, 4 application
assignments, and 4 research-related assignments.
Procedure
At the beginning of the first class meeting, all students were
asked to complete the ASIRA. Student names were randomly
assigned to the treatment group (Random Distribution Applications) or the control group (Normal Distribution Applications). The original sample of students included 58 students;
however, four students dropped out of the class within the
first four weeks of the semester, leaving a final sample of

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4

K. H. LARWIN AND D. A. LARWIN

54 students for the analyses. Students in the control and
experimental groups were asked to complete their specific
assignments independently, as an out-of-class homework assignment. Other than the experimental group’s exposure to
the learning modules, and the different assignments, students
in both groups received the same instruction, and completed
the same activities in and out of class, and completed the
same quizzes each week of the semester.
The treatment group participants were given four assignments that were based on peer-reviewed research projects.
These peer-reviewed research projects were a part of learning modules developed by Rossman et al. (2008). Each one
of the four assignments included two sections of Rossman
et al.’s learning modules. These lessons asked students to
review a brief discussion of a research project, and guided
students through an application intended to develop and reinforce student’s understanding of statistical inference.
The first assignment consisted of two mini-lessons that
developed the ideas of random distribution theory using a
study by Antonioli and Reveley (2005); the second assignment also consisted of two mini-lessons that were used to
further reinforce the ideas of random distribution theory, and
expanded to consideration of binomial distributions (using
data from Hamlin, Wynn, & Bloom, 2007). The third assignment also consisted of two mini-lessons and focused on
random distribution with binomial models (using data from
Todorov, Mandisodza, Goren, & Hall, 2005); and the fourth
assignment incorporated bootstrapping procedures to further
develop the idea of statistical inference (using data from
Stickgold, James, & Hobson, 2000). For each of these assignments, students read a brief description of a study, and
then completed resampling procedures using playing cards
or poker chips. Students responded to a few questions regarding their findings from their initial resampling with the poker
chips and playing cards. Students were then asked to use
online applets that would repeat the resampling procedures
upwards of 1000 times, and answer questions regarding their
findings, based on the larger samples. On each exam, students
in the treatment group were asked to respond to four questions about each of the research studies and applets, which
they completed.
The control group participants were also given four assignments over the course of the semester in which they
were to review and critique peer-reviewed published research
projects. These assignments were estimated to require the
same amount of time and the same amount of writing as
the assignments given to the treatment group. Unlike the
shortened research project summaries read by the treatment
group as part of the learning modules, the control group participants’ assignments required that they read and respond
to an entire research article. These papers incorporated the
four primary topic areas being discussed during that particular section of the course when the project was assigned.
The four assignments included a paper by Richardson and
Aguiar (2004); reporting only descriptive statistics, a study

by Materia et al. (2005) incorporating correlation analyses;
a study including regression analyses by Troisi, Christopher,
and Marek (2006); and a paper by Luskin, Aberman, and
DeLorenzo (2006) including both a t-test analysis and analysis of variance (ANOVA). In a two-page response, students
were asked to summarize the papers, discuss the sample and
sampling procedures, the use of the data, and identify variables and whether the variables were used appropriately by
the researchers. Students were to defend their response with
information from the class text and class notes and discussion. On each exam, students in the control group condition
were asked to respond to four questions about each of the
research studies that they reviewed.
The ASIRA was administered to the students during the
first class meeting, and again during the last class meeting, in
an effort to assess students’ inferential and reasoning skills.
This assessment was taken at the beginning of the first and last
class sessions, and was administered as a computer-adapted
assessment.

RESULTS
The data indicated that the control group performed slightly
better on the pretest (M = 42.04, SD = 13.53) than did the
treatment group (M = 37.59, SD = 11.17). The results from
the pretest were examined in an effort to assess whether there
were any pre-existing differences in the two study groups.
An independent sample t test indicated that there were no
significant differences in the pretest ASIRA scores of the
students (n = 54) from the two groups, t(52) = 1.32, p =
.193. This data is presented in Table 1.
At the end of the semester, the ASIRA was readministered to the same students (n = 54). A repeated measures
ANOVA was conducted and results indicated significant differences in students’ scores from pre- to posttest, F(1, 52) =
45.94, p < .001, partial η2 = .469, with the pretest scores
being significantly lower (M = 39.81, SD = 12.49) than the
posttest scores (M = 53.15, SD = 13.81). Additionally, significant differences were revealed between groups from preto posttest, F(1, 52) = 17.56, p < .001, partial η2 = .248.
The data are presented in Table 2.
Specifically, students in the control group showed significantly lower posttest scores on the ASIRA (M = 47.22, SD =
13.25) relative to students in the treatment condition (M =
59.07, SD = 11.85). An independent sample t test examining
TABLE 1
Pretest Means and Standard Deviations, by Group
Control
Measure
Pretest

Experimental

M

SD

M

SD

t

42.04

13.53

37.59

11.17

1.32 (52)

RANDOM DISTRIBUTION THEORY
TABLE 2
Within- and Between-Groups Results
Source

df

Within groups
Between groups
Total

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∗∗∗ p

MS

1
1
52

F
∗∗∗

4800.00
1792.59
104.469

45.94
∗∗∗
17.56
—

< .001.

final grade point average by group also revealed significant
differences, t(52) = 4.35, p < .001; the control group ended
the semester with significantly fewer points (M = 277.94,
SD = 70.04) relative to the treatment group (M = 342.27,
SD = 31.79) based on a maximum of 400 total points. This
data is presented in Table 3.

DISCUSSION
The present study demonstrated the effectiveness of pedagogical tools designed to introduce random distribution theory
concepts to undergraduate business statistics students. Relative to students in the control group who were exposed to
teaching exercises based on traditional statistics instructional
techniques employing normal distribution theory concepts,
students in the experimental group exposed to teaching techniques based on random distribution theory concepts demonstrated greater mastery of course material as measured by
final course point totals, and demonstrated superior gains on
pre- and posttest measures of statistical inference and reasoning skills. These findings are consistent with a growing
consensus among statistical researchers and educators that
statistics instruction would prosper from a shift in focus from
normal distribution theory to random distribution theory. The
present study represents the first empirical demonstration of
the veracity of these assertions.
One possible limitation to the current investigation is the
difference between the assignments for the experimental and
control groups. The learning modules were originally written with three sections in each module. The decision was
made to assign only two sections of the learning modules
for each assignment, in an effort to make the time necesTABLE 3
Final Grade Point Average for Students in the Control
Group Relative to Experimental Group
Control

Course points (out of
400 possible points)
∗∗∗ p

< .001.

Experimental

M

SD

M

SD

t

277.94

70.04

342.27

31.79

4.35
∗∗∗
(52)

5

sary to complete the assignment equivalent to the control
group assignments. Students from prior semesters reported
that the completion time for one section of the simulation
assignment was approximately 30–45 min; students reported
spending approximately 1 hr on their article assignments. Another possible criticism of the experimental conditions of the
present study might concern the fun factor of the simulation
assignments relative to the article assignments. It is arguable
that the greater fun and enjoyment associated with the experimental group exercises was responsible for the gains in the
experimental group relative to the control group, rather than
the shift in focus to the random distribution theory-based
techniques. However, the control group articles were chosen
because of their relevance to the students’ lives and the potential tie-in with the business students’ other coursework.
Although reading and responding to an article may not be
as entertaining for the student as working through activities
with a computerized simulator, these articles do offer the
student an opportunity to apply the knowledge covered in
required course matter.
At the 2005 U.S. Conference on Teaching Statistics, Cobb
(2005) insisted that statistics educators should stop focusing
on classical methods of approximation, such as t tests and
F tests, and introduce the concepts of statistical inference
with simulations and randomization. Cobb posited that with
the present availability of computer technology, the time has
come to leave behind the teaching of outdated approximation
procedures based on assumptions of normal distribution theory. He maintained that the logic of inference should be introduced through randomization and bootstrapping techniques
through which students can observe randomization procedures and how statistical conclusions come about through
these randomization simulations.
In the present study we sought to incorporate the cooperative learning approaches, heavily supported in the literature, with the random distribution applications suggested
by Cobb (2007). Although 30 years of research suggests
that constructivist approaches to teaching statistics should
improve student attitudes and learning, research continues
to suggest that these approaches are not in fact alleviating
the negative feelings shared by an overwhelming majority of
students surveyed regarding their statistics education (e.g.,
Onwuegbuzie, 2004). The results of the present investigation suggest that students can benefit from an introduction
to the ideas of statistical inference based on randomization
and bootstrapping techniques. In addition to the much supported constructivist pedagogy, random distribution simulations, such as those developed by Rossman et al. (2008),
can help to improve student perceptions regarding the difficulty and relevance of statistics to their future business
careers.
According to Johnson and John (2003), the aim of business statistics coursework is to develop statistical thinking
skills that help students to understand and interpret data.
They maintained that in order to achieve this objective

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K. H. LARWIN AND D. A. LARWIN

students must be engaged not only in the course information, but actively work at discovering the meaning of data,
the importance and relevance of statistical concepts, and they
must be actively involved in constructing an understanding of
the influence of data where different distributions, different
sample sizes, and differing degrees of variability are at play.
The random distribution theory approach to teaching and
learning statistics provides business professors with a tool
that can engage students to construct an understanding of
statistics at this level. Teaching traditional estimation approaches may result in students who understand the subtle
nuances of abstract statistical theory, but who also continue
to struggle to actually use statistical methods to analyze data
effectively. One student involved in the present study, as a
participant in the experimental group, who had previously
completed an AP statistics course in high school and an introductory level statistics class in his first year in college,
indicated in an email: “although I have had statistics in the
past, and did very well in those classes, I did not realize until
now what I did not understand. The idea of ‘significantly
different than what would be expected to occur by chance’
had no real meaning until I completed these simulation assignments. So, thank you. I thought I might be bored in this
class, but now I know that I really do understand the concepts
that are foundational to statistics.”
Today’s business student is interested in more than the traditional talk and chalk lecture, coupled with well-meaning
assignments, followed by midterm and final exams. Activities using random distribution theory, such as those presented
here, can engage and intrigue students, and enhance their
understanding and their performance in their present coursework and enhance their ability to think statistically in their
future professional and academic careers.

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Zanakis, S. H., & Valenzi, E. (1997). Student anxiety and attitudes in business statistics. Journal of Education for Business, 73, 10–16.

APPENDIX A

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Assessment of Statistical Inference and Reasoning Ability
1. A small object was weighed on the same scale separately by nine students in a science class. The weight
(in grams) recorded by each student was as follows:
6.2, 6.0, 6.0, 15.3, 6.1, 6.3, 6.2, 6.15, 6.2. The students
want to determine as accurately as they can the actual weight of this object. Of the following methods,
which would you recommend they use.

A. Use the most common number, which is 6.2.

B. Use the 6.15 since it is the most accurate
weighting.

C. Add up the nine numbers, and divide by nine.

D. Throw out the 15.3, and add up the 8 numbers
and divide by 8.
Answer: D
2. The following message is printed on a bottle of prescription medicine: WARNING—For applications to
skin areas there is a 15% chance of developing a rash.
If a rash develops, consult your physician. Which of
the following options is the best interpretation of this
warning?

A. Don’t use this medication on your skin; there’s
a good chance of developing a rash.

B. For application to the skin, apply only 15% of
the recommended dose.

C. If a rash develops, it will probably only involve
15% of the skin.

D. About 15 of 100 people who use this medication develop a rash.

E. There is hardly a chance of getting a rash using
this medication.
Answer: D
3. A teacher wants to change the seating arrangement in
her class in the hope that it will increase the number
of comments her students make. She first decides
to see how many comments students make with the
current seating arrangement. A record of the number
of comments made by her eight students during one
class period was as follows: 0, 5, 2, 22, 3, 2, 1, 2. The
teacher wants to summarize this data by computing
the typical number of comments made that day. Of

7

the following methods, which would you recommend
she use?

A. Use the most common number, which is 2.

B. Add up the eight numbers and divide by eight.

C. Throw out the 22, add up the remaining seven
numbers and divide by seven.

D. Throw out the zero, add up the remaining
seven numbers and divide by seven.
Answer: A
4. The Springfield Meteorological Center wanted to determine the accuracy of their weather forecasts. They
searched their record for those days when the forecaster had reported a 70% chance of rain. They compared these forecasts to records of whether or not it
actually rained on those particular days. The forecast
of 70% chance of rain can be considered very accurate
if it rained on:

A. 95%-100% of the days.

B. 85%-94% of the days.

C. 75%-84% of the days.

D. 65%-74% of the days.

E. 55%-64% of the days.
Answer: D
5. Two containers, labeled A and B, are filled with red
and blue marbles in the following quantities: Container A has 6 red and 4 blue; Container B has
60 red and 40 blue. Each container is shaken vigorously. After choosing one of the containers, you
will reach in and, without looking, draw out a marble. If the marble is blue, you win $50. Which container gives you the best chance of drawing a blue
marble?

A. Container A

B. Container B

C. Equal chance from each container
Answer: C
6. Which of the following sequences is most likely to
result from flipping a fair coin 5 times?

A. HHHTT

B. THHTH

C. THTTT

D. HTHTH

E. All four sequences are equally likely
Answer: E
7. Listed below are the same sequences of Heads and
Tails that were listed in item eight. Which of the
sequences is least likely to occur?

A. HHHTT

B. THHTH

8

K. H. LARWIN AND D. A. LARWIN


C. THTTT

D. HTHTH

E. All four sequences are equally unlikely.
Answer: E

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8. Half of the newborns are girls and half are boys.
Hospital A records an average of 50 births a day.
Hospital B records an average of 10 births a day.
On a particular day, which hospital is more likely to
record 80% or more female births?

A. Hospital A

B. Hospital B

C. The two hospitals are equally likely to report
such an event.
Answer: B
9. The school committee of a small town wanted to
determine the average number of children per household in their town. They divided the total number of
children in the town by 50, the total number of households. Which of the following statements must be true
if the average children per household is 2.2?

A. Half the households in the town have more
than two children.

B. More households in the town have three children than have two children.

C. There are a total of 110 children in the town.

D. There are 2.2 children in the town for every
adult.

E. The most common number of children in a
household is 2.

F. None of the above
Answer: C
10. When two dice are simultaneously thrown it is possible that one of the following two results occurs: (1) A
5 and a 6 are obtained or (2) a five is obtained twice.
Select the most valid response:

A. The chances of obtaining each of these results
are equal.

B. There is more chance of obtaining result (1).

C. There is more chance of obtaining result (2).

D. It is impossible to give an answer.
Answer: B
11. When three dice are simultaneously thrown, which of
the following is most likely to be obtained?

A. A 5, a 3, and a 6

B. A five three times

C. A five twice and a 3

D. All three results are equally likely.

Answer: A
12. When three dice are simultaneously thrown, which of
these three results is least likely to occur?

A. A 5, a 3, and a 6

B. A five three times

C. A five twice, and a three

D. All three results are equally likely.
Answer: B
13. For one month, 500 elementary students kept a
daily record of the hours spent watching television.
The average number of hours reported was 28. The
researcher conducting the study also obtained report cards for each of the students. They found
that the students who did well in school spent less
time watching television than those students who
did poorly. Listed below are several responses regarding the results of this research. Which is most
valid?

A. The sample size of 500 is too small to permit
drawing conclusions.

B. If a student decreased the amount of time spent
watching television, his or her performance in
school would improve.

C. Even though students who did well watched
less television, this doesn’t necessarily mean
that watching television hurt school performance.

D. One month is not long enough period of time
to estimate how many hours the student really
spend watching television.

E. The research demonstrates that watching television causes poorer performance in school.

F. I don’t agree with any of these statements.
Answer: C
14. You are investigating a claim that men are more likely
than women to snore. You take a random sample of
men and a random sample of women and ask whether
they snore (according to family members). If the difference in the proportions (who snore) between the
two groups turns out not to be statistically significant, which of the following is the best conclusion to
draw?

A. You have found strong evidence that there is
no difference between the groups.

B. You have not found enough evidence to conclude that there is a difference between the
groups.

C. Because the result is not significant, the study
does not support any conclusion.

RANDOM DISTRIBUTION THEORY

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Answer: B

9

Answer: C

15. If the difference in the proportions (who snore) between the two groups does turn out to be statistically significant, which of the following is a valid
conclusion?

18. Reconsider the previous question. Now think about
not possible explanations but ‘plausible’ explanations. Which is the more plausible explanation for
the result?


A. It would not be surprising to obtain the observed sample results is there is really no difference between men and women.

B. It would be very surprising to obtain the observed sample results if there
is really no difference between men and
women.

C. It would be very surprising to obtain the observed sample results if there is really a difference between men and women.


A. Men and women do not differ on this issue
but there is a small chance that random sampling alone led to the difference we observed
between the two groups.

B. Men and women differ on this issue.

C. These explanations are equally plausible.

Answer: B
16. Suppose that the difference between the sample
groups turns out not to be significant, even though
your review of the research suggested that there really is a difference between men and women. Which
conclusion is reasonable?

A. Something went wrong with the analysis.

B. There must not be a difference after all.

C. The sample size might have been too
small.
Answer: C
17. If the difference in the proportions (who snore) between the two groups does turn out to be statistically
significant, which of the following is a possible explanation for this result?

A. Men and women do not differ on this issue
but there is a small chance that random sampling alone led to the difference we observed
between the two groups.

B. Men and women differ on this issue.

C. Either of the other explanations is possible.

Answer: B
19. Suppose that two different studies are conducted on
this issue. Study A finds that 40 our of 100 women
sampled report snoring, compared to 20 of 100 men.
Study B finds that 35 of 10 women report snoring,
compared to 25 of 100 men. Which study provides
stronger evidence that there is a difference between
men and women on this issue?

A. Study A

B. Study B

C. The strength of evidence would be similar for
these two studies.
Answer: A
20. Suppose two more studies are conducted on this issue. Both studies find that 30% of women sampled
report snoring, compared to 20% of men. But Study
C consists of 100 people of each gender, while Study
D consists of 40 people of each gender. Which study
provides stronger evidence that there is a difference
between men and women on this issue.

A. Study C

B. Study D

C. The strength of evidence would be similar for
these two studies.
Answer: A