Goodness of Fit Test

Used to check whether a set of observed categorical data follows a theoretical distribution such as uniform or Poisson.

Hypotheses

$H_0$: Data follow the specified distribution.

$H_1$: Data do not follow the specified distribution.

Test Statistic

Here:

• $O_i$: observed frequency for class $i$
• $E_i$: expected frequency for class $i$ under $H_0$
• $k$: number of non-empty classes

Degrees of Freedom

Here:

• $p$: number of parameters estimated from the data

Decision Rule

Reject $H_0$ if $\chi^2 > \chi^2_{\nu, \alpha}$ (right-tail test).

Small Expected Frequencies

When any $E_i < 5$, merge adjacent classes until all $E_i \geq 5$. Recalculate $k$ and $\nu$ after merging.