We statisticians have an opportunity to help our nation regain a leadership role in international markets, but we will have to change the way we teach statistics before we can do our part. I have some suggestions for changes in how we teach Statistics 101.
The "problem of three prisoners", a counterintuitive teaser, is analyzed. It is representative of a class of probability puzzles where the correct solution depends on explication of underlying assumptions. Spontaneous beliefs concerning the problem and intuitive heuristics are reviewed. The psychological background of these beliefs is explored. Several attempts to find a simple criterion to predict whether and how the probability of the target event will change as a result of obtaining evidence are examined. However, despite the psychological appeal of these attempts, none proves to be valid in general. A necessary and sufficient condition for change in the probability of the target event, following observation of new data, is proposed. That criterion is an extension of the likelihood-ratio principle (which holds in the case of only two complementary alternatives) to any number of alternatives. Some didactic implications concerning the significance of the chance set-up and reliance on analogies are discussed.
Rather than talking about interesting students I have met over the year, this talk centers instead on how we can interest students in statistics. Two possible aspects of this issue are: a. how we can get more students to take one or more courses in statistics, b. how we can get more students to go into graduate studies in statistics.
This article describes a classroom activity designed to stimulate students to think creatively about methods of selecting a representative sample from a population. Students are presented with a data set consisting of gender, SAT verbal score, SAT mathematics score, and high school grade point average for 317 freshmen from North Carolina State University. The students, who have not yet studied sampling, work in groups of three or four to generate three possible methods for selecting a representative sample of 20 freshmen from the population of 317. Each group uses its proposed methods to select three samples and computes various summary statistics and plots for the variables in each sample. The students are then given corresponding information for the entire population. After comparing sample statistics and population parameters, the groups evaluate the advantages and disadvantages of the proposed sampling methods. During the two semesters that I have used this activity in my Statistics 101 class, students have "invented" simple random sampling, systematic sampling, stratified sampling, and various combinations thereof.
In 1982, at Mount Holyoke College, a group of faculty began to plan what was eventually to find its way into the course catalog as Interdepartmental 100 -- Case Studies in Quantitative Reasoning. This paper is about the QR course, given in five sections. First, I shall describe the structure of the QR course, emphasizing its mechanics and content: then I'll turn to a closer look at some of what makes the course distinctive for some of us who have taught it. I'll end with two sections on assessment and one on exportability, in the hope not only that our blueprints can help interested others avoid reinventing a wheel or two, but equally important, that our experience can also save others from reinventing some of our more spectacular flat tires.
In this article we have provided teachers with four examples of short run illustrations which students can analyze and reflect upon. The examples provided can be handled by high school students once they realize that with suitable assumptions, the binomial distribution provides a reasonable approximation to the operation of chance in reality.
Sixty-one children, from 4 to 11 years old, were presented with two sets, each containing blue and yellow elements. Each time, one colour was pointed out as the payoff colour (POC). The child had to choose the set from which he or she would draw at random a POC element in order to be rewarded. The sets were of varying sizes with different proportions of the two colours. The problem was to select the higher of the two probabilities. Three kinds of materials were used; Pairs of urns with blue and yellow beads, pairs of roulettes divided into blue and yellow sectors, and pairs of spinning tops, likewise divided into two colours. Roughly around the age of six, children started to select the greater of the two probabilities systematically. The dominant error was selecting the set with the greater number of POC elements. Verbal concepts of probability and chance were explored and some 'egocentric' thought processes were described. The study indicates that probability concepts could be introduced into school teaching even in the first grades. The deterministic orientation in the instruction for young ages should be attenuated, permitting concepts of uncertainty right from the beginning.
This paper describes recent work carried out by the Schools Council Project on Statistical Education. The Project advocates a problem-solving approach towards the teaching of statistics in secondary schools (11-16 years of age range). It sees statistics as an interdisciplinary subject primarily concerned with data. The article illustrates these important aspects with teaching materials which have been developed and tested extensively in a variety of schools. Finally an evaluation and assessment of the project's work is presented.
The present investigation adopted a debiasing approach to the judgmental error known as the conjunction fallacy. Such an approach was used to determine the extent to which the conjunction fallacy reflects task specific misunderstanding of particular judgment problems. The results suggest that (a) subjects' misunderstanding of conjunction problems is indeed somewhat task specific, and (b) a debiasing approach can effectively lower but not eliminate the conjunctive error rate for problems that do not strongly implicate representativeness thinking. Educational strategies based on statistical and probabilistic knowledge are discussed as an approach to debiasing inferential errors like the conjunction fallacy.
This article is concerned with a recent debate on the generality of utility theory. It has been argued by Lopes that decisions regarding preferences between gambles are different for unique and repeated gambles. The present article provides empirical support for the need to distinguish between these two. It is proposed that violations of utility theory obtained under unique conditions, cannot necessarily be generalized to repeated conditions.