Goodness of Fit¶

Chi-Square Distributions¶

A continuous probability distribution of the sum of the squares of \(k\) independent variables

Each of these independent variables follows a standard normal distribution

• Positively skewed
• Non-negative
• dof = \(k-1\)
• \(\uparrow\) dof \(\implies\) chi-square distribution closer to normal distribution

When Is Chi-Square Distribution Used¶

• Goodness of fit tests
• Tests of independence
• Tests of homogeneity

Hypothesis Tests for Goodness of Fit¶

A measure of how well real-life observed data fits a theoretical model

Calculate Chi-Square Value¶

\(\chi^2 = \sum \dfrac{(observed - expected)^2}{expected}\)

Conclude A Hypothesis Test¶

• \(\chi^2 > critical\ value \implies\) reject \(H_0\)
• \(\chi^2 < critical\ value \implies\) not reject \(H_0\)