Segment 19. The Chi Square Statistic

From Computational Statistics (CSE383M and CS395T)
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To Calculate

1. Prove the assertion on lecture slide 5, namely that, for a multivariate normal distribution, the quantity <math>({\mathbf x-\mathbf\mu})^T{\mathbf\Sigma}^{-1}({\mathbf x-\mathbf\mu})</math>, where <math>\mathbf x</math> is a random draw from the multivariate normal, is <math>\chi^2</math> distributed.

To Think About

1. Why are we so interested in t-values? Why do we square them?

2. Suppose you measure a bunch of quantities <math>x_i</math>, each of which is measured with a measurement accuracy <math>\sigma_i</math> and has a theoretically expected value <math>\mu_i</math>. Describe in detail how you might use a chi-square test statistic as a p-value test to see if your theory is viable? Should your test be 1 or 2 tailed?