Sums and Products of Jointly Distributed Random Variables: A Simplified Approach

Sheldon H. Stein

Cleveland State University
Journal of Statistics Education Volume 13, Number 3 (2005),

Abstract

Three basic theorems concerning expected values and variances of sums and products of random variables play an important role in mathematical statistics and its applications in education, business, the social sciences, and the natural sciences. A solid understanding of these theorems requires that students be familiar with the proofs of these theorems. But while students who major in mathematics and other technical fields should have no difficulties coping with these proofs, students who major in education, business, and the social sciences often find it difficult to follow these proofs. In many textbooks and courses in statistics which are geared to the latter group, mathematical proofs are sometimes omitted because students find the mathematics too confusing. In this paper, we present a simpler approach to these proofs. This paper will be useful for those who teach students whose level of mathematical maturity does not include a solid grasp of differential calculus.

Keywords: Covariance; Joint probability distribution; Means; Variances.

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Teaching Introductory Statistics Online – Satisfying the Students

Gail E. Tudor

Husson College

Abstract

This paper describes the components of a successful, online, introductory statistics course and shares students’ comments and evaluations of each component. Past studies have shown that quality interaction with the professor is lacking in many online courses. While students want a course that is well organized and easy to follow, they also want to interact with the professor and other students. Interactions in this course took place through small group discussions, emails, weekly announcements and graded exams. The course also contained lecture slides with audio prepared by the professor. As the variety and quantity of interaction increased, student satisfaction with the amount of interaction with the professor increased from 75% the first year of the course to 99% the fifth year. Overall satisfaction with the online course increased from 93% the first year to 100% the fifth year.

Keywords: Course design; Online versus traditional learning; Statistics education.

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Attitudes Toward Statistics and Their Relationship with Short- and Long-Term Exam Results

Stijn Vanhoof

Ana Elisa Castro Sotos

Patrick Onghena

Lieven Verschaffel

Wim Van Dooren

Wim Van den Noortgate

Katholieke Universiteit Leuven
Journal of Statistics Education Volume 14, Number 3 (2006),

Abstract

This study uses the Attitudes Toward Statistics (ATS) scale (Wise 1985) to investigate the attitudes toward statistics and the relationship of those attitudes with short- and long-term statistics exam results for university students taking statistics courses in a five year Educational Sciences curriculum. Compared to the findings from previous studies, the results indicate that the sample of undergraduate students have relatively negative attitudes toward the use of statistics in their field of study but relatively positive attitudes toward the course of statistics in which they are enrolled. Similar to other studies, we find a relationship between the attitudes toward the course and the results on the first year statistics exam. Additionally, we investigate the relationship between the attitudes and the long-term exam results. A positive relationship is found between students’ attitudes toward the use of statistics in their field of study and the dissertation grade. This relationship does not differ systematically from the one between the first year statistics exam results and the dissertation grade in the fifth year. Thus, the affective and cognitive measures at the beginning of the curriculum are equally predictive for long-term exam results. Finally, this study reveals that the relationship between attitudes toward statistics and exam results is content-specific: We do not find a relationship between attitudes and general exam results, only between attitudes and results on statistics exams.

Keywords: Assessment; Attitudes Toward Statistics scale.

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Classroom Research: Assessment of Student Understanding of Sampling Distributions of Means and the Central Limit Theorem in Post-Calculus Probability

M. Leigh Lunsford - Longwood University
Ginger Holmes Rowell - Middle Tennessee State University
Tracy Goodson-Espy - Appalachian State University

Abstract

We applied a classroom research model to investigate student understanding of sampling distributions of sample means and the Central Limit Theorem in post-calculus introductory probability and statistics courses. Using a quantitative assessment tool developed by previous researchers and a qualitative assessment tool developed by the authors, we embarked on data exploration of our students’ responses on these assessments. We observed various trends regarding their understanding of the concepts including results that were consistent with research completed previously (by other authors) for algebra-based introductory level statistics students. We also used the information obtained from our data exploration and our experiences in the classroom to examine and conjecture about possible reasons for our results.

Keywords: Action Research.

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An Active Tutorial on Distance Sampling

Alice Richardson - University of Canberra

Abstract

The technique of distance sampling is widely used to monitor biological populations. This paper documents an in-class activity to introduce students to the concepts and the mechanics of distance sampling in a simple situation that is relevant to their own experiences. Preparation details are described. Variations and extensions to the activity are also suggested.

Keywords: Estimation; Proportions; Sampling distribution; Statistical education.

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Probability in Action: the Red Traffic Light

John A. Shanks University of Otago
Journal of Statistics Education Volume 15, Number 1 (2007),

Abstract

Emphasis on problem solving in mathematics has gained considerable attention in recent years. While statistics teaching has always been problem driven, the same cannot be said for the teaching of probability where discrete examples involving coins and playing cards are often the norm. This article describes an application of simple probability distributions to a practical problem involving a car’s approach to a red traffic light, and draws on the ideas of density functions, expected value and conditional distributions. It provides a valuable exercise in applying theory in a practical context.

Keywords: Distributions; Modelling; Optimization; Problem solving.

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A Bubble Mixture Experiment Project for Use in an Advanced Design of Experiments Class

Stefan H. Steiner - University of Waterloo
Michael Hamada - Los Alamos National Laboratory
Bethany J. Giddings White - University of Waterloo
Vadim Kutsyy - Guardian Analytics
Sofia Mosesova - University of Waterloo
Geoffrey Salloum - Camosun College

Journal of Statistics Education Volume 15, Number 1 (2007)

Abstract

This article gives an example of how student-conducted experiments can enhance a course in the design of experiments. We focus on a project whose aim is to find a good mixture of water, soap and glycerin for making soap bubbles. This project is relatively straightforward to implement and understand. At its most basic level the project introduces students to mixture experiments and general issues in experimental design such as choosing and measuring an appropriate response, selecting a design, the effect of using repeats versus replicates, model building, making predictions, etc. To accommodate more advanced students, the project can be easily enhanced to draw on various areas of statistics, such as generalized linear models, robust design, and optimal design. Therefore it is ideal for a graduate level course as it encourages students to look beyond the basics presented in class.

Keywords: Constrained experimental region; Generalized linear model; Optimal design; Poisson regression; Robust parameter design.

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Rank-order Conjoint Experiments: efficiency and design

by Vermeulen B; Goos P; Vandebroek M.

Abstract
In a rank-order conjoint experiment, the respondent is asked to rank a number of alternatives instead of choosing the preferred one, as is the standard procedure in conjoint choice experiments. In this paper, we study the efficiency of those experiments and propose a D-optimality criterion for rank-order conjoint experiments to find designs yielding the most precise parameter estimators. For that purpose, an expression of the Fisher information matrix for the rank-ordered multinomial logit model is derived which clearly shows how much additional information is provided by each extra ranking step made by the respondent. A simulation study shows that Bayesian D-optimal ranking designs are slightly better than Bayesian D-optimal choice designs and (near-)orthogonal designs and perform considerably better than other commonly used designs in marketing in terms of estimation and prediction accuracy. Finally, it is shown that improvements of about 50% to 60% in estimation and prediction accuracy can be obtained by ranking a second alternative. If the respondent ranks a third alternative, a further improvement of 30%in estimation and prediction accuracy is obtained.

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Using R in Introductory Statistics Courses with the pmg Graphical User Interface

John Verzani CUNY / College of Staten Island
Journal of Statistics Education Volume 16, Number 1 (2008), www.amstat.org/publications/jse/v16n1/verzani.html

Abstract
The pmg add-on package for the open source statistics software R is described. This package provides a simple to use graphical user interface (GUI) that allows introductory statistics students, without advanced computing skills, to quickly create the graphical and numeric summaries expected of them.

Keywords: Statistics software; Statistical computing; R; Introductory statistics; EDA; Exploratory data analysis.

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Collaboration in Learning and Teaching Statistics

Cary J. Roseth, Joan B. Garfield, and Dani Ben-Zvi Michigan State University, USA,

University of Minnesota, Twin Cities, USA, University of Haifa, Haifa, Israel
Journal of Statistics Education Volume 16, Number 1 (2008), www.amstat.org/publications/jse/v16n1/roseth.html
Copyright © 2008 by Cary J. Roseth, Joan B. Garfield, and Dani Ben-Zvi all rights reserved.

Abstract
This paper provides practical examples of how statistics educators may apply a cooperative framework to classroom teaching and teacher collaboration. Building on the premise that statistics instruction ought to resemble statistical practice, an inherently cooperative enterprise, our purpose is to highlight specific ways in which cooperative methods may translate to statistics education. So doing, we hope to address the concerns of those statistics educators who are reluctant to adopt more student-centered teaching strategies, as well as those educators who have tried these methods but ultimately returned to more traditional, teacher-centered instruction.

Keywords: Collaboration; Cooperative learning; Collaborative teaching; Statistics education.

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A Comparison of Computer-Assisted Instruction and the Traditional Method of Teaching Basic Statistics

Carmelita Y. Ragasa University of the East Manila
Journal of Statistics Education Volume 16, Number 1 (2008), www.amstat.org/publications/jse/v16n1/ragasa.html
Copyright © 2008 by Carmelita Y. Ragasa all rights reserved.

Abstract.
The objective of the study is to determine if there is a significant difference in the effects of the treatment and control groups on achievement as well as on attitude as measured by the posttest. A class of 38 sophomore college students in the basic statistics taught with the use of computer-assisted instruction and another class of 15 students with the use of the traditional method from the University of the East, Manila (SY 2003-2004) were the focus of this study. The research method used was the quasi-experimental, non-equivalent control group design. The statistical tool was the Multiple Analysis of Covariance. The researcher made use of the CD-ROM prepared by Math Advantage (1997) to serve as the teaching medium for the experimental group. The following summarizes the findings of the study. The achievement posttest of the treatment group has higher estimated marginal means than the control group and it is reversed in the attitude posttest. Using Hotelling’s Trace for the multivariate test, the achievement pretest, attitude pretest, and the two groups have a significant effect on the dependent variables, achievement posttest and attitude posttest. Using covariates to control for the effects of additional variables that might affect performance the attitude pretest accounts for about 56% of the variability in the two groups while achievement pretest about 15%. Levene’s test shows that the homogeneity of variances assumption between the two groups is met for achievement posttest but not for attitude posttest. The univariate effects for achievement posttest that are significant are achievement pretest, college entrance test overall score, and groups. The univariate effects that are significant for attitude posttest are attitude pretest and high school general weighted average.

Keywords: Descriptive statistics; Multimedia; Learning.

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Reasoning and Communicating in the Language of Statistics

Carol S. Parke Duquesne University
Journal of Statistics Education Volume 16, Number 1 (2008), www.amstat.org/publications/jse/v16n1/parke.html
Copyright © 2008 by Carol S. Parke all rights reserved.

Abstract
Although graduate students in education are frequently required to write papers throughout their coursework, they typically have limited experience in communicating in the language of statistics, both verbally and in written form. To succeed in their future careers, students must be provided with opportunities to develop deep understandings of concepts, develop reasoning skills, and become familiar with verbalizing and writing about statistics. The instructional approach described here spans the entire semester of a statistics course and consists of several aspects including cognitively rich individual assignments, small group activities, and a student-led scoring activity. To demonstrate the impact of this approach on student learning, qualitative and quantitative data were collected from students in two statistics courses. Several assessments indicate improvement in students’ reasoning and understanding, written and verbal communication, and confidence.

Keywords: Conceptual understanding; Confidence; Interpretation of results; Verbal communication; Written communication.

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A Photographic View of Cumulative Distribution Functions

Robert W. Jernigan American University, Washington, DC
Journal of Statistics Education Volume 16, Number 1 (2008), www.amstat.org/publications/jse/v16n1/jernigan.html
Copyright © 2008 by Robert W. Jernigan all rights reserved.

Abstract
This article shows a concrete and easy recognizable view of a cumulative distribution function(cdf). Photograph views of the search tabs on dictionaries are used to increase students’ understanding and facility with the concept of a cumulative distribution function. Projects for student investigations are also given. This motivation and view helps the cdf become a bit more tangible and understandable.

Keywords: Alphabet; Kolmogorov; Probability distribution function(pdf); Scrabble; Smirnov; Student projects.

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Keeping it real, keeping them interested and keeping it in their minds

Peter P. Howley The University of Newcastle
Journal of Statistics Education Volume 16, Number 1 (2008), www.amstat.org/publications/jse/v16n1/howley.html
Copyright © 2008 by Peter P. Howley all rights reserved.

Abstract.
As part of many universities’ Business degrees, students will undertake an introductory statistics course. Lecturers need to help these students appreciate and recognise the value of possessing quantitative skills and to learn and apply such skills. Three components to teaching that address these aims as well as the interdependence of these components as part of a process which enhances the teaching environment and student outcomes are described. Methods and examples to perform the techniques and ideas are provided along with a discussion of their implementation and effectiveness after delivery in a large first year course.

Keywords: Teaching introductory statistics; Pedagogy; Increasing student confidence; Improved learning methods; Teaching materials.

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The Lure of Statistics in Data Mining

Lovleen Kumar Grover and Rajni Mehra Guru Nanak Dev University, India,

BBK DAV College for Women, India
Journal of Statistics Education Volume 16, Number 1 (2008), www.amstat.org/publications/jse/v16n1/grover.html
Copyright © 2008 by Lovleen Kumar Grover and Rajni Mehra all rights reserved.

Abstract
The field of Data Mining like Statistics concerns itself with "learning from data" or "turning data into information". For statisticians the term "Data mining" has a pejorative meaning. Instead of finding useful patterns in large volumes of data as in the case of Statistics, data mining has the connotation of searching for data to fit preconceived ideas. Here we try to discuss the similarities and differences as well as the relationships between statisticians and data miners. This article is intended to bridge some of the gap between the people of these two communities.

Keywords: Censored data; Databases; Data dredging; Data fishing; Data mining; Exploratory data analysis; Knowledge discover in data mining; Truncated data.

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The Effect of a Student-Designed Data Collection Project on Attitudes Toward Statistics

Lisa J. Carnell High Point University
Journal of Statistics Education Volume 16, Number 1 (2008), www.amstat.org/publications/jse/v16n1/carnell.html
Copyright © 2008 by Lisa J. Carnell all rights reserved.

Abstract
Students often enter an introductory statistics class with less than positive attitudes about the subject. They tend to believe statistics is difficult and irrelevant to their lives. Observational evidence from previous studies suggests including projects in a statistics course may enhance students’ attitudes toward statistics. This study examines the relationship between inclusion of a student-designed data collection project in an introductory statistics course and 6 components comprising students’ attitudes toward statistics. The sample consisted of 42 college students enrolled in an introductory statistics course. Comparisons of those who completed the student-designed data collection project (n = 24) and those who did not complete the project (n = 18) suggest that inclusion of a project may not significantly impact students’ attitudes toward statistics. However, these findings must be viewed as only a preliminary step in the study of the effect of projects on attitudes toward statistics.

Keywords: Attitudinal scale; Statistics education; Project-based learning; Comparative study.

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Application of Transformations in Parametric Inference

by Naomi Brownstein and Marianna Pensky University of Central Florida Journal of Statistics Education Volume 16, Number 1 (2008), www.amstat.org/publications/jse/v16n1/brownstein.html
Copyright © 2008 by Naomi Brownstein and Marianna Pensky all rights reserved.

Abstract
The objective of the present paper is to provide a simple approach to statistical inference using the method of transformations of variables. We demonstrate performance of this powerful tool on examples of constructions of various estimation procedures, hypothesis testing, Bayes analysis and statistical inference for the stress-strength systems. We argue that the tool of transformations not only should be used more widely in statistical research but should become a routine part of calculus-based courses of statistics. Finally, we provide sample problems for such a course as well as possible undergraduate reserach projects which utilize transformations of variables.

Keywords: Transformations of variables; Estimation; Testing; Stress-strength model; Bayesian inference.

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An Active Tutorial on Distance Sampling

Alice Richardson University of Canberra
Journal of Statistics Education Volume 15, Number 1 (2007)

Abstract
The technique of distance sampling is widely used to monitor biological populations. This paper documents an in-class activity to introduce students to the concepts and the mechanics of distance sampling in a simple situation that is relevant to their own experiences. Preparation details are described. Variations and extensions to the activity are also suggested.

Key Words:Estimation; Proportions; Sampling distribution; Statistical education.

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Trashball: A Logistic Regression Classroom Activity

Christopher H. Morrell and Richard E. Auer Loyola College in Maryland
Journal of Statistics Education Volume 15, Number 1 (2007)

Abstract
In the early 1990's, the National Science Foundation funded many research projects for improving statistical education. Many of these stressed the need for classroom activities that illustrate important issues of designing experiments, generating quality data, fitting models, and performing statistical tests. Our paper describes such an activity on logistic regression that is useful in second applied statistics courses. The activity involves students attempting to toss a ball into a trash can from various distances. The outcome is whether or not students are successful in tossing the ball into the trash can. This activity and the adjoining homework assignments illustrate the binary nature of a response variable, fitting and interpreting simple and multiple logistic regression models, and the use of odds and odds ratios.

Key Words: Odds, Odds ratio; Problem solving

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The Availability Heuristic: A Redux

Laurie H. Rubel Brooklyn College of the City University of New York
Journal of Statistics Education Volume 15, Number 2 (2007)

Abstract
This article reports on a subset of results from a larger study which examined middle and high school students’ probabilistic reasoning. Students in grades 5, 7, 9, and 11 at a boys’ school (n=173) completed a Probability Inventory, which required students to answer and justify their responses to ten items. Supplemental clinical interviews were conducted with 33 of the students. This article describes students’ specific reasoning strategies to a task familiar from the literature (Tversky and Kahneman, 1973). The results call into question the dominance of the availability heuristic among school students and present other frameworks of student reasoning.

Keywords: availability heuristic, combinatorial thinking, middle school, high school

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