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Volume 13 (2005)

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An International Journal on the Teaching and Learning of Statistics

JSE Volume 13, Number 2 Abstracts

Matthew A. Carlton
Pedigrees, Prizes, and Prisoners: The Misuse of Conditional Probability

We present and discuss three examples of misapplication of the notion of conditional probability. In each example, we present the problem along with a published and/or well-known incorrect - but seemingly plausible - solution. We then give a careful treatment of the correct solution, in large part to show how careful application of basic probability rules can help students to spot and avoid these mistakes. With each example, we also hope to illustrate the importance of having students draw a tree diagram and/or a sample space for probability problems not involving data (i.e., where a contingency table might not be obviously applicable).

Key Words: Bayes’ Rule; Monty Hall Problem; Pedigree analysis; Prisoner’s Paradox.

Christopher J. Mecklin and Robert G. Donnelly
Powerball, Expected Value, and the Law of (very) Large Numbers

In this paper, we consider some combinatorial and statistical aspects of the popular “Powerball” lottery game. It is not difficult for students in an introductory statistics course to compute the probabilities of winning various prizes, including the “jackpot” in the Powerball game. Assuming a unique jackpot winner, it is not difficult to find the expected value and the variance of the probability distribution for the dollar prize amount. In certain circumstances, the expected value is positive, which might suggest that it would be desirable to buy Powerball tickets. However, due to the extremely high coefficient of variation in this problem, we use the law of large numbers to show that we would need to buy an untenable number of tickets to be reasonably confident of making a profit. We also consider the impact of sharing the jackpot with other winners.

Key Words: Coefficient of variation; Expectation; Lottery; Probability.

Paul T. von Hippel
Mean, Median, and Skew: Correcting a Textbook Rule

Many textbooks teach a rule of thumb stating that the mean is right of the median under right skew, and left of the median under left skew. This rule fails with surprising frequency. It can fail in multimodal distributions, or in distributions where one tail is long but the other is heavy. Most commonly, though, the rule fails in discrete distributions where the areas to the left and right of the median are not equal. Such distributions not only contradict the textbook relationship between mean, median, and skew, they also contradict the textbook interpretation of the median. We discuss ways to correct ideas about mean, median, and skew, while enhancing the desired intuition.

Key Words: Asymmetry; Central tendency; Extreme values; Influence; Mean-median-mode inequality; Mode; Outliers; Robustness; Sensitivity.

Scott J. Seipel and Charles H. Apigian
Perfectionism in Students: Implications in the Instruction of Statistics

Previous research has linked perfectionism to anxiety in the statistics classroom and academic performance in general. This article investigates the impact of the individual components of perfectionism on academic performance of students in the statistics classroom. The results of this research show a clear positive relationship between a student’s personal standards and academic performance consistent with the literature. Surprisingly, the inherent need of some students for organization and structure was found to be negatively related to academic performance. This finding suggests that the organization of statistics as perceived by some students may not always foster understanding, resulting in student confusion and lack of achievement. This infers that statistics instructors may need to put sufficient emphasis on the underlying composition of statistical ideas and the linking of statistical techniques that are presented in the classroom and in the textbook. The implications of these results are discussed in terms of current trends in the reform of the statistics curriculum and approaches that may improve the clarity of the underlying structure of statistics.

Key Words: Business Statistics; Introductory Statistics Course; Perfectionism; Students.

H. Dean Johnson and Nairanjana Dasgupta
Traditional versus Non-traditional Teaching: Perspectives of Students in Introductory Statistics Classes

Although there has been a considerable amount of work evaluating the effects of different (non-traditional) instructional styles, inquiries into students’ preferences of instructional style have been few. From 1998-2001, we surveyed introductory statistics students regarding various aspects of their class preferences, especially the teaching style they prefer. We analyzed the data for the purpose of seeing if there has been an increasing trend in preference towards non-traditional methods. Our results are inconclusive (p = 0.35) about the presence of such a trend. However, the overall proportion of students preferring non-traditional classes is higher than students preferring traditional classes (p < 0.001). We also used the survey data to investigate the possible attributes that relate to preference. Using Stepwise Logistic regression (with = 0.10) we find that the students’ ideal class-size, the number of years since they graduated from high school, the perceived learning styles of the students, and the attitudes of students towards the use of visual aids and hands-on activities are all significantly related to the teaching style preferences of students.

Key Words: Binary logistic regression; Cochran-Armitrage trend test; Pedagogical Research.

Datasets and Stories

Michael Kahn
An Exhalent Problem for Teaching Statistics

A dataset concerning the relationship between respiratory function (measured by forced expiratory volume, FEV) and smoking provides a powerful tool for investigating a wide variety of statistical matters. This paper gives a brief description of the problem, the data, and several issues and analyses suggested by the problem of quantifying the relationship between FEV and smoking.

Key Words: Forced expiratory volume; Semester-long discussions; Statistics education.

Terence C. Mills
Predicting Body Fat Using Data on the BMI

A data set contained in the Journal of Statistical Education’s data archive provides a way of exploring regression analysis at a variety of teaching levels. An appropriate functional form for the relationship between percentage body fat and the BMI is shown to be the semi-logarithmic, with variation in the BMI accounting for a little over half of the variation in body fat. The fairly modest strength of the relationship implies that confidence intervals for body fat, and tolerance intervals for BMI, can be quite wide, so that strict reliance on the BMI as a measure of body fat, and hence obesity, is unwarranted. Nevertheless, when fitting percentage body fat as a function of the class of “power weight for height indices”, i.e., indices of the form weight/heightp, the BMI, with a height exponent of p = 2, is an appropriate choice to make.

Key Words: Body Mass Index; Functional form; Prediction; Regression.

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