A Concrete Strategy for Teaching Hypothesis Testing

Article excerpt

1. INTRODUCTION

This paper presents a simple device (hereafter referred to as the demonstrator) that can be used as a teaching aid for introducing the basic concepts in the "classical" approaches to hypothesis testing in a coherent way. The demonstrator can be used to illustrate Fisher's (1935, 1970) procedure of testing null hypotheses and Neyman-Pearson's (1928, 1933) procedure where one.is forced to choose between two rival hypotheses and the concept of power is introduced. The demonstrator is not intended to illustrate the Bayesian inferential procedures (Lindley 1965) nor the decision theory approach originated by Wald (1950).

2. DIFFICULTIES IN UNDERSTANDING THE SUBJECT

In my experience over many years of teaching an introductory course in statistics attended by undergraduate students in the behavioral sciences at the University of Leuven, almost all difficulties with the comprehension of the theory of hypothesis testing result from the following two points:

1. The fact that the subject is traditionally approached in an interweaved way from three perspectives:

a. the perspective of the ignorant statistician who does not know "the state of nature" (the real world) and can only hypothesize and reason about it in a conditional form,

b. the perspective of the state of nature, and

c. the perspective of the student who is studying the subject, but who is at the same time also an omniscient observer because he is informed about the state of nature and the way of operating of the statistician.

2. The usual instructional practice of putting distributions referring to different cases (i.e., "[H.sub.0] is true" and "[H.sub.0] is false") on the same set of axes. For most students this combined presentation conflicts with the knowledge that only one of the distributions fits the case in any concrete situation. A related confusing practice is to omit explicit plotting of the distribution representing the state of nature, and to refer to it by the distribution used for testing a true hypothesis. Hence the conceptual difference between a distribution that refers to the real world and one that refers to a hypothetical construction is faded.

In order to avoid confusion of perspectives and ambiguity in graphical representations, the present instructional method proposes a three-level frame of reference in conducting hypothesis testing. Separate sets of axes are reserved for plotting: (1) the sampling distribution in the state of nature, (2) the sampling distribution representing the case where [H.sub.0] is true, and (3) the sampling distribution representing the case where [H.sub.0] is false.

The three-level frame is based on the conviction that the common practice to represent distributions under different states on the same set of axes, although being perfectly appropriate for treating the problem from the point of view of the statistician who cannot distinguish at any stage between testing a true or false hypothesis, is ill-suited for educational purposes. Students are always informed about the state of nature. Consequently, they spontaneously contrast testing a true versus a false hypothesis. Hence it seems natural to use a representational format that fits in with this mental approach by assigning separate sets of axes to distributions under different states. The demonstrator was designed to materialize this idea.

Of course, the simple use of the demonstrator does not lead to insight automatically. The accompanying verbal commentary that will be outlined in Section 4 is equally important. Because the theory of hypothesis testing can be found in all introductory textbooks, the commentary in this paper will be limited to specifying the way in which the concepts are presented in dialogue with the apparatus.

3. DESCRIPTION OF THE APPARATUS

The demonstrator [ILLUSTRATION FOR FIGURE 1 OMITTED] is simple and inexpensive to construct. (A copy of the demonstrator can be purchased at production cost (approximately $150) from the author. …