Hypothesis Testing and Chi-Square Tests

Chapter 1: Introduction

When comparing variables across groups, we examine whether the proportions of a variable are similar across two or more groups.
The null hypothesis (H_0) assumes the proportions are similar across all groups.
The alternative hypothesis (H_a) states that at least one of the proportions is different.
Rejecting H_0 implies that one or more of the proportions are not the same.

If you "fail to reject" the null hypothesis, it means you don't have enough statistical proof to say that any of the proportions are wrong.
Test of Independence:
- The null hypothesis (H_0) is that two variables are independent.
- The alternative hypothesis (H_a) is that the two variables are not independent (they are associated).
- When conducting a Chi-square test, you calculate a p-value.
- Example: If the p-value is 0.1, and the significance level is lower than that (e.g. 0.05),
  you fail to reject the null hypothesis (H_0).
- Failing to reject H_0 means there's not enough statistical evidence to prove the variables are not independent.

To calculate the Chi-square statistic (\chi^2), you need observed and expected values.
Degrees of freedom depend on the specific test being conducted.

For the Goodness of Fit test:
- Degrees of freedom = (number of categories - 1).
- Example: If a variable (e.g., color) has five categories, the degrees of freedom is 4.
For Homogeneity and Test of Independence (two-way table):
- Degrees of freedom = (number of categories in variable 1 - 1) * (number of categories in variable 2 - 1).
- Example: If one variable has two categories (yes/no) and another variable has three categories, the degrees of freedom = (2-1) * (3-1) = 1 * 2 = 2.
For multiple groups (two or three) and one variable:
- Degrees of freedom = (number of categories - 1).