Samples, Sampling Distributions, & the Central Limit Theorem

Examples for class exercises ranged from simple approaches such as the use of imaginary bags of marbles (Zerbolio, 1989), wooden blocks (Wybraniec & Wilmoth, 1999), pages from the phone book (Woolley, 1998), or cards with numbers written on them (Couch, 1987; Johnson, 1986) to more sophisticated demonstrations involving commuter train performance (Porkess, 2003), Keno (Johnson, 2006), and M & M's (Dyck & Gee, 1998).

Introduction to Null Hypothesis Testing

Seier and Robe (2002) provided students with a worksheet describing the case of a biology student seeking to determine whether ducks prefer green food versus another color. The worksheet provided students with background information on null and alternative hypotheses, p-values, hypothesis testing, and error rates. In addition, a series of questions on the worksheet required that students apply their knowledge of inferential statistics. Maret and Ziemba (1997) introduced another example from biology for teaching hypothesis testing—the relationship between declining leopard frog populations and the introduction of non-native green sunfish. Hodgson, Andersen, Robinson-Cox, and Jones (2004) took advantage of the Student and Teachers as Researchers (STAR) program, which examines water quality issues, to teach theory and practical applications of inferential statistics. Specifically, they assigned student streams and provided them with simulated samples based on existing data to assess whether the proportion of macroinvertebrate species (e.g., shrimp, crayfish, caddisflies) differed as a function of pollution. Hodgson et al. recommended that instructors consider a field-based follow-up. Although these examples may be more appropriate to the biological sciences, they can easily be adapted for use with psychology students.

Violating ANOVA Assumptions

In addition to providing students with an understanding of the importance of the ANOVA assumptions, instructors also have the opportunity to educate students as to the best means to test these violations. For example, Berry, Mielke, and Mielke (2002) presented data from a soil-based lead test to illustrate the differences among the various methods of testing heterogeneity of variance. They compared the Gosset t test, Welch t test, Log transformed t test, Wilcoxon-Mann-Whitney test, and the Fisher-Pitman test to determine which was most accurate. Berry et al. concluded that the Fisher-Pitman test was the best approach to use given the fact it is "distribution free, data-dependent, and does not depend on the assumption of equal variances" (p. 500).

General Linear Model

Saville and Wood (1986) detailed a method for teaching statistics, using Euclidean N-dimensional space, which they felt would bridge the gap between the simple computational cookbook approach (i.e., sum of squares) and the sophisticated mathematical approach they deemed too complex for most undergraduates. The authors provided the necessary algebraic and geometric concepts, proofs, and several analysis of variance and regression examples. They asserted that the methods they presented provide a "unified and comprehensive" approach to the teaching of ANOVA, regression, and ANCOVA at an elementary level (p. 213).

However, Cobb (1984) stated that the use of Euclidean geometry, although a more parsimonious approach, would nonetheless be difficult for the average student in the social sciences to comprehend. Consequently, he proposed an algorithmic alternative to teaching ANOVA that focused primarily on the linear decomposition of the data through a series of numerical adjustments followed by the more traditional sum of squares approach. By introducing the ANOVA in this fashion, Cobb asserted that students would be more likely to see the connections between the sum of squares and the data. Consequently, he felt that this approach would be "easier to learn, understand, and recognize as a single unified method that adapts to a great variety of data sets" (p. 120). To assist instructors, Cobb provided a general course guide, examples, and a computer program to assist with the calculations.

Varner and Utts (1976) presented a regression model as an alternative to the analysis of variance for testing interactions. In addition to introducing students to the general linear model, this approach allows for the use of continuous data versus necessitating the use of categorical data—as is the case with the ANOVA. The authors provided classroom presentation suggestions, statistical notation, and SPSS exercises. Gaito and Shermer (1985) encouraged instructors to use expected mean squares [E(MS)] when teaching the analysis of variance as a means to bridge the simplicity of the cookbook approach and more mathematically sophisticated approaches such as the Euclidian geometry. According to the authors, a student exposed to this approach is "capable of operating with greater understanding and is more alert to the attacking of novel designs" (p. 515). They also found that students taught using E(MS) are more successful, excited, and more willing to extend their knowledge to other designs with minimal assistance.

To assist students’ understanding of the relationship between the analysis of variance and regression, Eisenhauer (2006) recommended that instructors gather published television information (e.g., TV guide) regarding the age of movies shown on a Saturday during a specific month. During class, students compared the average age of movies shown on satellite versus non-satellite channels using a t-test—and confirmed the results using an ANOVA. Then, to demonstrate the applicability of the general linear model to this problem, Eisenhauer combined the two samples, introduced a dummy variable (0 = satellite; 1 = non-satellite), and had the class conduct a regression analysis on the data. The exercise could also be extended to include multiple regression by including a third station (e.g., HBO). Eisenhauer reported that exercise drew upon earlier concepts (e.g., hypothesis testing), integrated material from other portions of the class, and enabled students to develop a more sophisticated understanding of regression equations, so that they were "not quite so mysterious" (p. 80).

Computer Applications

Several researchers have used Java applets to facilitate student understanding of the central limit theorem (e.g., West & Ogden, 1998), confidence intervals (e.g., Bertie & Farrington, 2003; West & Ogden, 1998) power (e.g., Aberson et al., 2002; Anderson-Cook & Dorai-Raj, 2003; West & Ogden, 1998), and ANOVA (e.g., Strum-Beiss, 2005).

References

Anderson-Cook, C. M., & Dorai-Raj, S. (2003). Making the concepts of power and sample size relevant and accessible to students in introductory statistics courses using applets. Journal of Statistics Education, 11(3). Retrieved July 31, 2007, from http://www.amstat.org/publications/jse/v11n3/anderson-cook.html

Benjamin, L. T., Jr., & Lowman, K. D. (Eds.). (1981). Activities handbook for the teaching of psychology (Vol. 1). Washington DC: American Psychological Association.

Benjamin, L. T., Jr., Nodine, B. F., Ernst, R. M., & Broeker, C. B. (Eds.). (1999). Activities handbook for the teaching of psychology (Vol. 4). Washington, DC: American Psychological Association.

Bertie, A., & Farrington, P. (2003). Teaching confidence intervals with Java applets. Teaching Statistics, 25, 70-74.

Couch, J. M. (1987). Sampling fluctuation demonstration. In V. P. Makosky, L. G. Whittemore, & A. M. Rogers (Eds.), Activities handbook for the teaching of psychology (Vol. 2, pp. 201-203). Washington, DC: American Psychological Association.

Gourgey, A. F. (2000). A classroom simulation based on political polling to help students understand sampling distributions. Journal of Statistics Education, 8(3). Retrieved July 31, 2007, from http://www.amstat.org/publications/jse/secure/v8n3/gourgey.cfm

Hodgson, T., Anderson, L., Robinson-Cox, J., & Jones, C. (2004). Water quality statistics. Teaching Statistics, 26, 2-8.

Johnson, R. W. (2006). Optimal Keno strategies and the central limit theorem. Teaching Statistics, 28, 26-29.

Makosky, V. P., Sileo, C. C., Whittemore, L. G., Landry, C. P., & Skutley, M. L. (Eds.). (1990). Activities handbook for the teaching of psychology (Vol. 3). Washington, DC: American Psychological Association.

Makosky, V. P., Whittemore, L. G., & Rogers, A. M. (Eds.). (1987). Activities handbook for the teaching of psychology (Vol. 2). Washington, DC: American Psychological Association.

Porkess, R. (2003). Collecting data on train performance. Teaching Statistics, 25, 49-53.

Varner, L., & Utts, J. (1976). Parallel prediction lines: A test for interaction. Journal of Educational Research, 70, 63-66.

West, R. W., & Ogden, R. T. (1998). Interactive demonstrations for statistics education on the World Wide Web. Journal of Statistics Education, 6(3). Retrieved July 31, 2007, from http://www.amstat.org/publications/jse/v6n3/west.html

Woolley, T. W. (1998). A note on illustrating the central limit theorem. Teaching Statistics, 20, 89-90.