Limit Theorems

  • In these activities designed to introduce sampling distributions and the Central Limit Theorem, students generate several small samples and note patterns in the distributions of the means and proportions that they themselves calculate from these samples. Outside of class, students generate samples of dice rolls and coin spins and draw random samples from small populations for which data is given on each individual. Students report their sample means and proportions to the instructor who then compiles the results into a single data file for in-class exploration of sampling distributions and the Central Limit Theorem. Key words: Sampling distribution, sample mean, sample proportion, central limit theorem
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  • This applet allows the user to choose from several discrete distributions and to see what happens to the distribution of a sum of these random variables. The labels on the x-axis are misleading for underlying distributions, such as the geometric, which cannot be negative.
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  • This applet allows a person to add up to 50 points onto its green viewing screen. After each point is added by clicking on the screen with the mouse, a red line will appear. This red line represents a line passing through (Average x, Average y) with a slope that can be altered by clicking the Left or Right buttons. The slope of this line may also be changed by dragging the mouse either right or left. By clicking on Show Best Fit, a blue best fit line will be calculated by the computer.
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  • This article describes an interactive activity illustrating general properties of hypothesis testing and hypothesis tests for proportions. Students generate, collect, and analyze data. Through simulation, students explore hypothesis testing concepts. Concepts illustrated are: interpretation of p-values, type I error rate, type II error rate, power, and the relationship between type I and type II error rates and power. This activity is appropriate for use in an introductory college or high school statistics course. Key words: hypothesis test on a proportion, type I and II errors, power, p-values, simulation
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  • This activity leads students to appreciate the usefulness of simulations for approximating probabilities. It also provides them with experience calculating probabilities based on geometric arguments and using the bivariate normal distribution. We have used it in courses in probability and mathematical statistics, as well as in an introductory statistics course at the post-calculus level. Students are expected to approximate the solution through simulation before solving it exactly. They are also expected to employ graphical as well as algebraic problem-solving strategies, in addition to their simulation analyses. Finally, students are asked to explain intuitively why it makes sense for the probabilities to change as they do. Key words: simulation, probability, geometry, independence, bivariate normal distribution
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  • This course site contains a complete set of course materials, including lecture notes, assignments and exams. This subject is a computer-oriented introduction to probability and data analysis. It is designed to give students the knowledge and practical experience they need to interpret lab and field data. The MATLAB?? programming language is used to perform virtual experiments and to analyze...
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  • This applet demonstrates that even a "small" effect can be important under some circumstances. Applicants from two groups apply for a job. The user manipulates the mean and the cut-off score in order to see the effects the small changes has on the number of people hired in each group. The effects on the proportion of hired applicants from each group are displayed.(Requires a browser that supports Java).
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  • The applet allows users to sample from a normal distribution or from a uniform distribution. It shows the expected values and the observed values and computes the deviation. Then, a chi-square test shows if the deviations are significant for both the normal and uniform distributions.
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  • This applet simulates experiments using 2 x 2 contingency tables. You specify the population proportions and the sample size and examine the effects on the probability of rejecting the null hypothesis.
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  • This applet demonstrates how the reliability of X and Y affect various aspects of the regression of Y on X.
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