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The Chi-Square Test Statistic

The chi-square test statistic  is an overall measure of how close the observed frequencies are to the expected frequencies. It has the form

\begin{displaymath}\chi^2= \mbox{SUM} \frac{(\mbox{observed frequency - expected frequency})^2}
{\mbox{expected frequency}}.

The null hypothesis of independence is rejected if $\chi^2$ is large, because this means that observed frequencies and expected frequencies are far apart. The chi-square curve is used to judge whether the calculated test statistic is large enough. We reject H0 if the test statistic is large enough so that the area beyond it (under the chi-square curve with (r-1)(c-1) degrees of freedom) is less than .05.

\epsfig{, height=3in, width=4in}

The P-value is the area greater than $\chi^2$ under the chi-square curve with (r-1)(c-1) degrees of freedom.