Test of difference of two variences based on samples variances

Assuming equal variance in populations is important if we want to use simple formula for the difference of the means based on pooled variance instead of long formula of degree of freedom and diffrent variances.

https://demosophy.org/estimators-for-the-difference-of-the-means-of-two-populations/

to check the assumption of equality we consider:

H0: $\sigma_1^{2}= \sigma _2^{2}$

H1: $\sigma_1^{2}\neq \sigma _2^{2}$

according to Cochran’s theorem s² follows a scaled chi-square distribution:

$(n-1)\frac{s^2}{\sigma^2}\sim\chi^2_{n-1}.$

this means

${s}^{2}=\frac{{{\chi }^{2}}_{n-1}*{\sigma}^{2} }{n-1}$

therefore”

$\frac{{s_1}^{2}}{{s_2}^{2}}=\frac{{}\frac{{{\chi }^{2}}_{m-1}*{\sigma_1}^{2} }{m-1}}{\frac{{{\chi }^{2}}_{n-1}*{\sigma_2}^{2} }{n-1}}$

if null hypothesis is true

$\frac{{s_1}^{2}}{{s_2}^{2}}=\frac{{}\frac{{{\chi }^{2}}_{m-1} }{m-1}}{\frac{{{\chi }^{2}}_{n-1} }{n-1}}$

and we know that if

and

be independent variates distributed as chi-squared with

and

degrees of freedom.We Define a statistic

as the ratio of the dispersions of the two distributions

this means that

$\frac{{s_1}^{2}}{{s_2}^{2}}$

has a $F_{m-1,n-1}$ distribution.

Demosophy.org

Test of difference of two variences based on samples variances

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