Two-Way Tables: The Full Four-Step Test

State — in words

For a two-way table the hypotheses are written in words, not symbols. Independence: "$H_0$: there is no association between [variable 1] and [variable 2] in the population; $H_a$: there is an association." Homogeneity: "$H_0$: the distribution of [variable] is the same for all [groups]; $H_a$: it is not the same for all groups." Choose by asking how the data were collected — one sample with two variables, or several samples of one variable.

Plan — expected counts and conditions

Under $H_0$ every cell's expected count is

Conditions: random sample(s) or random assignment; $10\%$ condition for each sample; and all expected counts at least $5$. Say which condition each check satisfies and quote the smallest expected count.

Do and conclude

$\chi^2=\sum\frac{(\text{observed}-\text{expected})^2}{\text{expected}}$ with $df=(r-1)(c-1)$; the P-value is always the upper tail. Example: $50$ boys and $50$ girls asked whether they prefer morning classes; $30$ boys and $20$ girls say yes. Every expected count is $25$, each of the four cells contributes $(5^2)/25=1$, so $\chi^2=4{,}0$ with $df=1$ and P-value $\approx0{,}046$. Conclude in context at the stated $\alpha$, then do the follow-up graders reward: name the cells with the largest components and say in which direction the observed count departs from the expected — that is what "there is an association" actually means.