Demos and Applets
for
Introductory Statistics
Contents:
Venn Diagrams Demo by J Puranen
- Apart from their useful role in Set Theory, Venn Diagrams are also used when (uncertain) events in the study of Probability are formally represented as sets. Representing events as sets help us quantify their uncertainty and also facilitate the derivation of rules governing the calculation and manipulation of probabilities.
- This demo allows one to interactively visualize various combinations of three sets obtained by taking unions, intersections and complements. Point to A, B etc. with your mouse pointer and you will see the corresponding set shaded in the Venn diagram.
Normal Approximation to Binomial Demo
- The normal approximation to the binomial refers to the approximation of a probability computed using a binomial distribution by another probability computed using an appropriate normal distribution. Geometrically, this involves approximating a shaded area under a binomial probability histogram by a shaded area under a suitable approximating normal density.
- This demo illustrates the above geometrical idea. It also demonstrates how the approximation is improved by making a continuity correction.
- By definition, a confidence interval for a (scalar) parameter is a random interval which contains the parameter with a certain probability. Given some data, we can compute a particular realization of the confidence interval which may or may not contain the parameter of interest. If we repeat the interval construction procedure a large number of times, the high level of confidence usually attached to such intervals means that a large proportion of them will contain the true parameter in the long run.
- This applet demonstrates the above statements. Using simulated data, you can repeatedly generate sets of 50 confidence intervals for the mean at a given level of confidence. A diagram is given in which each interval is represented by a line segment. Since data are simulated, you know the true mean and therefore you can keep track on how many intervals actually contain the true mean. Intervals which do not contain the mean are represented by red line segments. By running the simulations repeatedly for a given alpha, you will find that the long run relative frequency of red line segments converge to the value of alpha set by you. You can also simulate intervals with higher levels of confidence by choosing smaller values for alpha.
- The correlation coefficient is a summary measure which quantifies the strength and direction of linear relationship between two variables measured on a continuous scale.
- This demo allows one to vary the sample correlation coefficient, r, and observe the correponding scatter plot for two variables. As you vary r from +1 to -1, you'll see a strong positive linear relationship in the scatter plot gradually changing to a random scatter (i.e., corresponding to r=0) which then changes gradually to a strong negative linear relationship.
- The simple linear regression model allows us to relate two variables statistically (rather than mathematically). What this means is that in this model, the expected value of the response variable, Y, is modelled as a linear function of an explanatory variable, x (e.g., average sales as a linear function of advertising expenditure). Due to random error, values measured for the pair (x,Y) do not fall exactly on the mean response line but scatter about it in a `linear fashion'.
- This is a multipurpose demo but is essentially concerned with how to fit a simple linear regression model given data from a bivariate sample. Press the Scatterplot, Formulas and Calculations buttons to see a plot of the data, the formulas used to calculate the regression parameters and step-by-step computation of these parameters, respectively. Also demonstrated is calculation of the sample correlation coefficient, r.