# Segment 19. The Chi Square Statistic

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### Problems

#### To Calculate

1. Prove the assertion on lecture slide 5, namely that, for a multivariate normal distribution, the quantity **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ({\mathbf x-\mathbf\mu})^T{\mathbf\Sigma}^{-1}({\mathbf x-\mathbf\mu})}**
, where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf x}**
is a random draw from the multivariate normal, is **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi^2}**
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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i}**
, each of which is measured with a measurement accuracy **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_i}**
and has a theoretically expected value **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_i}**
. 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?

#### Class Exercise

Data file: Media:mv_chi.txt