Independence: One sample, two categorical variables, exploring relationships within a single group. | Homogeneity: Two or more samples, one categorical variable, comparing distributions across groups.

Independence: Requires a simple random sample (SRS). | Homogeneity: Requires a stratified random sample OR randomly assigned treatments (for experiments).

Expected Count = (Row Total * Column Total) / Grand Total

$$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$$ where O is observed and E is expected.

df = (number of rows - 1) * (number of columns - 1)

When sampling without replacement, ensure the sample size (n) is less than 10% of the population size (N) to maintain independence: n < 0.10N.

All expected counts must be at least 5 to ensure the Chi-Square test is valid.

Random sampling ensures that the sample is representative of the population, which is a fundamental assumption for the validity of the test.

Contextual hypotheses clearly define the variables and populations being studied, making the results more meaningful and interpretable.

The p-value indicates the probability of observing a test statistic as extreme as, or more extreme than, the one calculated if the null hypothesis were true. It is used to make a decision about rejecting the null hypothesis.