Ivo D. Dinov, Nicolas Christou and Robert Gould
Law of Large Numbers: the Theory, Applications and Technology-based Education

Modern approaches for technology-based blended education utilize a variety of recently developed novel pedagogical, computational and network resources. Such attempts employ technology to deliver integrated, dynamically-linked, interactive-content and heterogeneous learning environments, which may improve student comprehension and information retention. In this paper, we describe one such innovative effort of using technological tools to expose students in probability and statistics courses to the theory, practice and usability of the Law of Large Numbers (LLN). We base our approach on integrating pedagogical instruments with the computational libraries developed by the Statistics Online Computational Resource (www.SOCR.ucla.edu). To achieve this merger we designed a new interactive Java applet and a corresponding demonstration activity that illustrate the concept and the applications of the LLN. The LLN applet and activity have common goals - to provide graphical representation of the LLN principle, build lasting student intuition and present the common misconceptions about the law of large numbers. Both the SOCR LLN applet and activity are freely available online to the community to test, validate and extend (Applet: http://socr.ucla.edu/htmls/exp/Coin_Toss_LLN_Experiment.html, and Activity: http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_Activities_LLN).

Key Words: Statistics education; Technology-based blended instruction; Applets; Law of large numbers; Limit theorems; SOCR.

Michael D. Ernst
Teaching Inference for Randomized Experiments

Nearly all introductory statistics textbooks include a chapter on data collection methods that includes a detailed discussion of both random sampling methods and randomized experiments. But when statistical inference is introduced in subsequent chapters, its justification is nearly always based on principles of random sampling methods. From the language and notation that is used to the conditions that students are told to check, there is usually no mention of randomized experiments until an example that is a randomized experiment is encountered, at which point the author(s) may offer a statement to the effect of "the randomization allows us to view the groups as independent random samples." But a good student (or even an average one) should ask, "Why?"

This paper shows, in a way easily accessible to students, why the usual inference procedures that are taught in an introductory course are often an appropriate approximation for randomized experiments even though the justification (the Central Limit Theorem) is based entirely on a random sampling model.

Key Words: ANOVA; Computing; Curriculum; Normal distribution; Randomization distribution; Randomization test; Sampling distribution; Two-sample t-test.

Jennifer J. Kaplan
Effect of Belief Bias on the Development of Undergraduate StudentsŐ Reasoning about Inference

Psychologists have discovered a phenomenon called "Belief Bias" in which subjects rate the strength of arguments based on the believability of the conclusions. This paper reports the results of a small qualitative pilot study of undergraduate students who had previously taken an algebra-based introduction to statistics class. The subjects in this study exhibited a form of Belief Bias when reasoning about statistical inference. In particular, the subjects in this study were more likely to question the experimental design of a study when they did not believe the conclusions reached by the study. While these results are based on a small sample, if replicated, the results have implications for the teaching of statistics. Specifically, when teaching hypothesis testing, statistics instructors should be mindful about the context of example problems used in class, make explicit links between inference to experimental design and actively engage students in discussions of both believability of conclusions and the types of arguments they find convincing.

Key Words: Statistics education; Statistical reasoning; Fundamental computational bias; Heuristics and biases.

Ilana Lavy and Michal Mashiach-Eizenberg
The Interplay Between Spoken Language and Informal Definitions of Statistical Concepts

Various terms are used to describe mathematical concepts, in general, and statistical concepts, in particular. Regarding statistical concepts in the Hebrew language, some of these terms have the same meaning both in their everyday use and in mathematics, such as Mode; some of them have a different meaning, such as Expected value and Life expectancy; and some have the opposite meaning, such as Significance level. Spoken language plays an important role in shaping how the informal statistical definitions taught in schools are remembered. In the present study we examine the impact of the everyday use of terms on the students' informal definitions of various statistical concepts. Though all the study participants were familiar with the concepts they were asked to define, a high percentage of them failed to provide correct definitions of the given statistical concepts. Analysis of the incorrect definitions revealed that the everyday use of the terms used to label the concepts, influenced the informal definitions provided by the students.

Key Words: Spoken language; Informal definition; Statistical concepts; Concept image; Concept definition; Dual nature of concepts.

John Marriott, Neville Davies and Liz Gibson
Teaching, Learning and Assessing Statistical Problem Solving

In this paper we report the results from a major UK government-funded project, started in 2005, to review statistics and handling data within the school mathematics curriculum for students up to age 16. As a result of a survey of teachers we developed new teaching materials that explicitly use a problem-solving approach for the teaching and learning of statistics through real contexts. We also report the development of a corresponding assessment regime and how this works in the classroom.

Controversially, in September 2006 the UK government announced that coursework was to be dropped for mathematics exams sat by 16-year-olds. A consequence of this decision is that areas of the curriculum previously only assessed via this method will no longer be assessed. These include the stages of design, collection of data, analysis and reporting which are essential components of a statistical investigation. The mechanism outlined here could provide some new and useful ways of coupling new teaching methods with learning and doing assessment - in short, they could go some way towards making up for the educational loss of not doing coursework. Also, our findings have implications for teaching, learning and assessing statistics for students of the subject at all ages.

Key Words: Problem solving; Teaching; Learning; Assessment.

Robert M. Pruzek and James E. Helmreich
Enhancing Dependent Sample Analyses with Graphics

A standard topic in many Introductory Statistics courses is the analysis of dependent samples. A simple graphical approach that is particularly relevant to dependent sample comparisons is presented, illustrated and discussed in the context of analyzing five real data sets. Each data set to be presented has been published in a textbook, usually introductory. Illustrations show that comprehensive graphical analyses often yield more nuanced, and sometimes quite different interpretations of data than are derived from standard numerical summaries. Indeed, several of our findings would not readily have been revealed without the aid of graphic or visual assessment. Several arguments made by John Tukey about data analysis are seen to have special force and relevance.

Key Words: Dependent samples; Graphical analyses; Matching; Blocking; Efficient designs; Repeated measures; R software; granova

Timothy J. Robinson, William A. Brenneman and William R. Myers
An Intuitive Graphical Approach to Understanding the Split-Plot Experiment

While split-plot designs have received considerable attention in the literature over the past decade, there seems to be a general lack of intuitive understanding of the error structure of these designs and the resulting statistical analysis. Typically, students learn the proper error terms for testing factors of a split-plot design via expected mean squares. This does not provide any true insight as far as why a particular error term is appropriate for a given factor effect. We provide a way to intuitively understand the error structure and resulting statistical analysis in split-plot designs through building on concepts found in simple designs, such as completely randomized and randomized complete block designs, and then provide a way for students to "see" the error structure graphically. The discussion is couched around an example from paper manufacturing.

Key Words: Hard to change factors; Restricted randomization; Whole-plot Factors; Sub-plot Factors.

Narelle Smith, Anna Reid and Peter Petocz
Representations of Internationalisation in Statistics Education

Internationalisation is an important but contentious issue in higher education. For some it means the facilitation of student mobility and an important source of funding for universities, while for others it forms a philosophy of teaching and student engagement, highlighting issues of global inequality. In this study, the papers from a recent statistics education conference, the 7th International Conference on Teaching Statistics, are subjected to a critical discourse analysis against a theoretical frame derived from research describing different ways of understanding and working with internationalisation. The analysis demonstrates how a specific discipline-based community - the statistics education community - involves itself with issues of internationalisation.

Key Words: Internationalisation; Statistics education; Critical discourse analysis; Phenomenography.

António Teixeira, Álvaro Rosa and Teresa Calapez
Statistical Power Analysis with Microsoft Excel: Normal Tests for One or Two Means as a Prelude to Using Non-Central Distributions to Calculate Power

This article presents statistical power analysis (SPA) based on the normal distribution using Excel, adopting textbook and SPA approaches. The objective is to present the latter in a comparative way within a framework that is familiar to textbook level readers, as a first step to understand SPA with other distributions. The analysis focuses on the case of the equality of the means of two populations with equal variances for independent samples with the same size.

This is the situation adopted as case 0 by Cohen (1988), a pioneer in the subject, to develop his set of tables and so, the present article can be seen as an introduction to Cohen's methodology applied to tests based on samples from normal populations. We also discuss how to extend the calculation to cases with other characteristics (cases 1 to 4), similarly to what Cohen proposes, as well as a brief discussion about the advantages and shortcomings of Excel. We teach mainly in the area of business and economics, which determines the scope of our analysis.

Key Words: Effect size; Excel; Non-central distributions; Non-centrality parameter; Normal distribution; Power.

Carla J. Thompson
Educational Statistics Authentic Learning CAPSULES: Community Action Projects for Students Utilizing Leadership and E-based Statistics

Since educational statistics is a core or general requirement of all students enrolled in graduate education programs, the need for high quality student engagement and appropriate authentic learning experiences is critical for promoting student interest and student success in the course. Based in authentic learning theory and engagement theory graduate educational statistics CAPSULES (Community Action Projects for Students Utilizing Leadership and E-based Statistics) engage graduate students in service-learning projects involving managing, conducting, and delivering authentic data-driven research. The community action projects utilizing leadership and e-based statistics skills are spearheaded by a university-based Community Outreach Research and Authentic Learning (CORAL) Center. The graduate educational statistics CAPSULES program includes: (1) restructuring educational statistics courses to include real-world active learning and authentic assessment; (2) providing opportunities for graduate students to engage in team-driven quantitative research prior to the thesis or dissertation experience with projects generated from community agencies/educational institutions; and (3) connecting graduate students with community action projects as research managers, leaders, and presenters. Highlights of initial formative and summative student outcomes are presented relative to specific examples of student-directed CAPSULES. Student outcomes from the CAPSULES program indicate positive increases in graduate students' attitudes toward statistics and research, and students' leadership and project management skills.

Key Words: Teaching graduate educational statistics; Community partnerships with higher education; Service learning; Authentic learning of statistics; Student engagement.

Marie Wiberg
Teaching Statistics in Integration with Psychology

The aim was to revise a statistics course in order to get the students motivated to learn statistics and to integrate statistics more throughout a psychology course. Further, we wish to make students become more interested in statistics and to help them see the importance of using statistics in psychology research. To achieve this goal, several changes were made in the course. The theoretical framework to motivate teaching method changes was taken from the statistics education literature together with the ideas of student-centered learning and Kolb's learning circle. One of the changes was to give the students research problems in the beginning of the course that were used throughout the course and which they should be able to solve at the end of the course. Other changes were to create a course webpage and to use more computer-based assignments instead of assignments with calculators. The students' test results and their answers on the Survey of Attitudes Toward Statistics, SATS, (Schau, Stevens, Dauphinee, & Del Vecchio, 1995) together with course evaluations showed that by changing the course structure and the teaching, students performed better, and were more positive towards statistics even though statistics was not their major.

Key Words: Student-centered learning, Research problems, Course revision.

Audbjorg Bjornsdottir and Joan Garfield
Statistics Education Articles from 2008

We located 58 articles that were published in 2008 that pertained to statistics education. In this column, we highlight a few of these articles that represent a variety of different journals that include statistics education in their focus. We also provide information about the journal and a link to their website so that abstracts of additional articles may be accessed and viewed.

Deborah J. Rumsey
Random Thoughts: Letting Go to Grow: Independent vs. Mutually Exclusive

Teachers often get caught up in the discussion of how to teach this concept or that concept, or how to explain this connection or that connection, but sometimes we should just stand back and be bold enough to ask the question, "Should we even be teaching this?"; "Is it really relevant to the modern statistics course?"; "Is it related to the GAISE guidelines?"; Do we ever use this idea again later in our course?" As we contemplate the future of teaching statistics, it's a good time to stop, think, and ask the hard questions. The theme of USCOTS 2009 (The United States Conference on Teaching Statistics) is "Letting Go to Grow". In that spirit I'd like to throw out some ideas regarding the classic 'independent vs. mutually exclusive' discussion that is still included in most introductory statistics textbooks and in many courses.

Constance H. McLaren and Concetta A. DePaolo
Movie Data

The Movie dataset contains weekend and daily per theater box office receipt data as well as total U.S. gross receipts for a set of 49 movies. Dates are provided for all time series values. The diverse list of movies was selected, not at random, but to spark student interest and to provide a range of box office values. The values provide a rich dataset to use for applications such as simple graphical analysis, a variety of time series and causal forecasting models, curve-fitting, and rate of change analysis. A series of assignment questions is included and the accompanying Instructor's Manual provides representative solutions.

Key Words: Time Series, Movie Box Office, Forecasting, Graphical Display of Data, Curve Fitting, Rate of Change