Chi-square !

Chi-Square Test of Independence:
- Determines whether two categorical variables are independent of each other.
- Example: Testing if gender and preference for a product are related.
Chi-Square Goodness-of-Fit Test:
- Compares the observed frequency distribution of a single categorical variable to an expected distribution.
- Example: Checking if the roll of a die is fair (expected equal probability for all outcomes).

The Chi-square statistic ( $\chi^2$ ) is calculated as:

$\chi^2 = \sum \frac{(O - E)^2}{E}$

Where:

State the hypotheses:
- Null hypothesis ( $H_0$ ): There is no association (independence) or the observed data fits the expected distribution.
- Alternative hypothesis ( $H_a$ ): There is an association (dependence) or the observed data does not fit the expected distribution.
Calculate expected frequencies:
- For independence: Use marginal totals.
- For goodness-of-fit: Based on theoretical proportions.
Compute the Chi-square statistic:
- Apply the formula to calculate $\chi^2$ .
Determine degrees of freedom ( $df$ ):
- $df = (r-1)(c-1)$ for independence tests, where $r$ and $c$ are the number of rows and columns in the contingency table.
- $df = \text{number of categories} - 1$ for goodness-of-fit tests.
Compare $\chi^2$ with the critical value:
- Use a Chi-square distribution table or p-value with the chosen significance level ( $\alpha$ ).
Make a decision:
- If $\chi^2$ is greater than the critical value or p-value < $\alpha$ , reject $H_0$ .

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