The ubiquitous angle

Previously we used the geometry of n-dimensional space to derive the paired samples t-test and its p-value. In the present paper we describe the 'ubiquitous' application of these results to single degree of freedom linear model hypothesis tests. As examples, we derive the p- and t-values for the independent samples t-test, for testing a contrast in an analysis of variance and for testing the slope in a simple linear regression analysis. An angle θ in n-dimensional space is again pivotal in the development of the ideas. The relationships between p, t, θ, F and the correlation coefficient are also described by using a 'statistical triangle'.