Visual Means Test

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The set of vectors in -space of norm 1 forms a sphere. Any particular vector , when rotated about the vector of means , defines a cone that intersects the sphere to form two spherical caps. The surface area of the caps divided by the surface area of the sphere equals the significance probability of the -test. Choose , the degrees of freedom; is then a vector in ()-space that has the -value corresponding to the other control. Although the sample vector lives in dimensions, a three-dimensional graphic is used to illustrate the cap-area phenomenon, with the cone angle determined by .


For testing the null hypothesis that the population mean is zero, the test statistic (where is the standard deviation of the sample vector) can be seen to be times the cotangent of the angle formed by and . The caps on the sphere and the corresponding areas in the distribution are colored red when the -value is 0.05 or less; otherwise it is green and the evidence against the null hypothesis is not strong.


Contributed by: Karl W. Heiner and Stan Wagon (August 2011)
(SUNY New Paltz and Macalester College)
Open content licensed under CC BY-NC-SA




[1] G. E. P. Box, W. G. Hunter, and J. S. Hunter, Statistics for Experimenters: An Introduction to Design, Data Analysis, and Model Building, 2nd ed., New York: Wiley–Interscience, 2005.

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