14.8: Formula Review

Test of Two Variances

\(\begin{array}{l}
H_0: \dfrac{\sigma_1^2}{\sigma_2^2}=\delta_0 \\
H_a: \dfrac{\sigma_1^2}{\sigma_2^2} \neq \delta_0
\end{array}\)

if \(\delta_0=1\) then

\(\begin{array}{l}
H_0: \sigma_1^2=\sigma_2^2 \\
H_a: \sigma_1^2 \neq \sigma_2
\end{array}\)

Test statistic is :

\(F_c=\dfrac{S_1^2}{S_2^2}\)

The F Distribution and the F-Ratio

\(\begin{array}{l}
S S_{\text {between }}=\sum\left[\dfrac{\left(s_j\right)^2}{n_j}\right]-\dfrac{\left(\sum s_j\right)^2}{n} \\
S S_{\text {total }}=\sum x^2-\dfrac{\left(\sum x\right)^2}{n} \\
S S_{\text {within }}=S S_{\text {total }}-S S_{\text {between }} \\
d f_{\text {between }}=d f(\text { num })=k-1 \\
d f_{\text {within }}=d f(\text { denom })=n-k \\
M S_{\text {between }}=\dfrac{S S_{\text {between }}}{d f_{\text {between }}} \\
M S_{\text {within }}=\dfrac{S S_{\text {within }}}{d f_{\text {within }}} \\
F=\dfrac{M S_{\text {between }}}{M S_{\text {within }}}
\end{array}\)

\(k=\) the number of groups
\(n_j=\) the size of the \(j^{t h}\) group
\(s_j=\) the sum of the values in the \(j^{\text {th }}\) group
\(n=\) the total number of all values (observations) combined
\(x=\) one value (one observation) from the data
\(s_{\bar{x}}{ }^2=\) the variance of the sample means
\(s^2{ }_{\text {pooled }}=\) the mean of the sample variances (pooled variance)