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June 5th, 2016, 11:58 AM   #1
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Determining Chi-Square test confidence

I tried this problem in the basic statistics forum but none there responded, so perhaps it's an advanced statistics problem. To set the stage, lets set up a typical sample problem:

A group of people are categorized by gender and also by political affiliation, and we wish to test the hypothesis that there is a relationship between the two categorizations.
\begin{array} {r|c c|c}
gender & republican & democrat & total\\
\hline
male & 30 & 20 & 50 \\
female & 20 & 30 & 50
\end{array}
\begin{array}{r l}
H_0 &: \text{gender and political affiliation are independent variables} \\
H_1 &: \text{gender and political affiliation are related} \\
\text{degrees of freedom} &: (rows - 1) \times (columns - 1) = 1
\end{array}
If the two dimensions were independent then we would have an expectation of equal portions in each of the four table entries
\begin{array} {r|c c|c}
gender & republican & democrat & total\\
\hline
male & 25 & 25 & 50 \\
female & 25 & 25 & 50
\end{array}
The test statistic $\chi^2$ is calculated by summing a computation for each cell of the table
$$\chi^2 = \sum{\frac{(observed - expected)^2}{expected}}=
\frac{(30-25)^2}{25} + \frac{(20-25)^2}{25} +
\frac{(20-25)^2}{25} + \frac{(30-25)^2}{25} = 4
$$

At this point we compare $\chi^2$ to a reference value $\chi^2_a$ drawn from a pre-calculated table for some confidence level $a$ in the appropriate number of degrees of freedom, and reject the null hypothesis $H_0$ if $\chi^2 > \chi^2_a$

Question
Is there a way to determine the value $a$ at which $\chi^2 = \chi^2_a$ ?
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June 5th, 2016, 09:53 PM   #2
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You need a computer to do that, though a would not be not be the confidence level any more - it would be the p-value. If you determine the confidence level after computing the chi-square statistic, this is data snooping and not good science, so you probably should not regard a as the confidence level.
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October 26th, 2017, 01:29 PM   #3
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The a value is just the p value of chi-square test here, I used this chi-square calculator to get a p value of 0.0455. Usually, the threshold is set at 0.05 or 0.01. For this chi-square test with a p value of 0.0455, it is statistically significant at p<0.05 level, but NOT at p<0.01 level.
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