Select the Null Hypothesis for a Test of Independence: H₀ and H₁

Answer

Select the statement that the two categorical variables are independent as the null hypothesis:

$$H_0:\ \text{The two categorical variables are independent.}$$

The alternative is that they are associated:

$$H_1:\ \text{The two categorical variables are not independent.}$$

The alternative is also commonly written $H_a$.

Translate independence into probabilities

Under $H_0$, knowing the category of one variable does not change the probability distribution of the other. For every row category $i$ and column category $j$,

$$P(A=i,B=j)=P(A=i)P(B=j).$$

For a contingency table, the expected count under this model is

$$E_{ij}=\frac{(\text{row }i\text{ total})(\text{column }j\text{ total})}{\text{grand total}}.$$

These expected counts need not be equal across cells. They reflect the observed margins under a model of independence.

Keep the conclusion separate from the hypothesis

A sufficiently small p-value provides evidence against independence. Failing to reject $H_0$ means the data do not provide sufficient evidence of association at the chosen significance level; it does not establish independence as a fact. An observed association also does not establish causation.

This question asks only for the hypotheses. Without observed counts and a significance level, there is no numerical test result to report.