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Consider the variables sex (male or female) and political party affiliation (Democrat or Republican). In this case, the null hypothesis and alternative hypothesis would be:

  • H0: Sex is independent from political party affiliation.
  • H1: Sex and political party affiliation are related.

I understand that a chi square test of independence can be used if I'm checking whether two categorical variables are dependent, so that's what I would use here.

However, what if I want to perform a test such as the below:

  • H0: Sex is independent from political party affiliation.
  • H1: Men are more likely to be democrats than women.

Chi square only tests for independence so I feel like it wouldn't provide enough information. But then what kind of test could be used here? How would one go about solving this problem?

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The key is that the first set of hypotheses yield a two-sided test whereas the second set of hypotheses yield a one-sided test.

The null hypothesis in scenario 2 is more correctly stated as "Men are more likely to be republican than women".

In a two-tailed test, you have to split your alpha between the two critical regions where a "significant" finding might be observed. Pearson's chi-square test of independence is not setup for one-sided testing, but doing so is simple: multiply alpha by two, recompute the critical region, check if the sample proportion favors males a democrats, and reject the null if the test statistic is in the critical region. This result should produce a one sided test of the correct size.

It may be useful to look up one and two sided test of differences in proportion using the normal approximation to the distribution of the sample proportion.

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  • $\begingroup$ My understanding is that since the chi square test relies on an asymmetrical distribution, it's not possible to perform a two tailed test with it. (See pictures on the wiki for reference: en.wikipedia.org/wiki/Chi-square_distribution) $\endgroup$
    – student
    Jun 1, 2021 at 21:37
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    $\begingroup$ @twix In my first year stats class, we covered one-tailed categorical tests. It's nothing to do with the asymmetry of the $\chi^2$ distribution, rather there's just the added nuance of actually looking at the direction of association. The normal approximation to proportion test is more intuitive, and you should probably use it. $\endgroup$
    – AdamO
    Jun 1, 2021 at 21:42
  • $\begingroup$ @twix, you can perform a one-tailed version of a z-test for the difference of proportions. $\endgroup$
    – gung - Reinstate Monica
    Jun 1, 2021 at 21:56

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