F distribution

$F$ distribution
- the ratio of two sample variances has an $F$ distribution
  - if both populations are normally distributed
- has 2 parameters
  - $ν_{1}$
    - degrees of freedom for the numerator
  - $ν_{2}$
    - degrees of freedom for the denominator
  - $\frac{S_{1}^{2}}{S_{2}^{2}} \sim F (n_{1} - 1, n_{2} - 1)$
    - (numerator df, denominator df)
- distribution of the F test statistic under $H_{0}$
  - if the null hypothesis is true, the value we get from F is contained in that distribution
    - extreme outlier $\to$ low p-value $\to$ low chance that our hypothesis is true
    - area to the right only, R