Segment 19: The Chi Square Statistic - 3/22/2012

From Computational Statistics (CSE383M and CS395T)
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Problem 1

Prove the assertion on lecture slide 5, namely that, for a multivariate normal distribution, the quantity , where is a random draw from the multivariate normal, is χ2 distributed.

To Think About 1

Why are we so interested in t-values? Why do we square them?
t-values are uni-normal random variables that help us represent the distance random draws are away from the supposed mean. I think we square them so we can see their distance from the mean as a positive number?

To Think About 2

Suppose you measure a bunch of quantities xi, each of which is measured with a measurement accuracy σi and has a theoretically expected value μi. 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?
I would sum up each <math>(\frac{x_i - \mu_i}{\sigma_i})^2</math> and plug that sum into a chi-square function in python. I would use a 2 sided test since extremes can happen on either side.