F-test and F-distribution When comparing variances, very often a F-test is used.
$$F=\frac{s_1^2}{s_2^2}$$
We then compare $F$ to an F-distribution. The assumption of this test is that the two samples (which variances are $s_1^2$ and $s_2^2$) are normally distributed.
Can you please explain why (given that the two samples are normally distributed) $\frac{s_1^2}{s_2^2}$ is F-distributed?
If I am not mistaken, if $U_1$ and $U_2$ are chi-squared distributed with degrees of freedom $df_1$ and $df_2$ respectively, then $\frac{U_1/df_1}{U_2/df_2}$ is F-distribution distributed. I would really appreciate an answer that also give some words between the link between the chi-squared distribution and the F-distribution.
 A: If we have a normal sample of size $n$, then the random variable $U_1=(n-1)s_1^2/\sigma^2_1$ follows a $\mathcal \chi^2_{(n-1)}$. This follows from the fact that a chi-square is a squared standard normal, and that $(n-1)s_1^2/\sigma^2_1$ is the sum of standardized normal random variables (write it out to see that).
Re-arrange and consider the random variable
$$s_1^2 = \sigma^2_1[U_1/(n-1)]$$
Now consider another normal sample of size $m$, and the associated magnitude
$$s_2^2 = \sigma^2_2[U_2/(m-1)]$$
We form the statistic
$$T = \frac {s_1^2}{s_2^2} = \frac{\sigma^2_1}{\sigma^2_2}\cdot \frac {[U_1/(n-1)]}{[U_2/(m-1)]}$$
The second term is a random variable that follows an $F$-distribution. So if the null hypothesis is true, $H_0 : \sigma^2_1 = \sigma^2_2$ the statistic will be a random variable following the $F$-distribution:
$$T|_{Η_0} =   \frac {[U_1/(n-1)]}{[U_2/(m-1)]} \sim F(n-1, m-1)$$
As for the link between the ratio of chi-squares and the F-distribution, is one of those "algebra of random variables" results that should make us very grateful to those people that actually devoted their time to derive them.
A: The variance is defined as
$V(X)=E((X-E(X))^2)$,
so for a standardnormal distribution we have
$V(X)=E(X^2)$ (since $\mu=0$).
This shows that the variance is $\chi^2$-distributed, because the $\chi^2$-distribution is defined as the distribution of (df-variate) $\sum X^2$, $X\sim N(0,1)$.
Since $F$ is as you defined the ratio of two normal variances, it is also the ratio of two $\chi^2$-variables.
This is how  the Normal, $\chi^2$, and $F$ distribution are linked.
