9.6: Chapter 12 Formula Review

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12.1 Test of Two Variances

\[H_{0} : \frac{\sigma_{1}^{2}}{\sigma_{2}^{2}}=\delta_{0}\nonumber\]

\[H_{a} : \frac{\sigma_{1}^{2}}{\sigma_{2}^{2}} \neq \delta_{0}\nonumber\]

if \(\delta_{0}=1\) then

\[H_{0} : \sigma_{1}^{2}=\sigma_{2}^{2}\nonumber\]

\[H_{a} : \sigma_{1}^{2} \neq \sigma_{2}\nonumber\]

Test statistic is :

\[F_{c}=\frac{S_{1}^{2}}{S_{2}^{2}}\nonumber\]

12.3 The F Distribution and the F-Ratio

\(S S_{\mathrm{between}}=\sum\left[\frac{\left(s_{j}\right)^{2}}{n_{j}}\right]-\frac{\left(\sum s_{j}\right)^{2}}{n}\)

\(S S_{\mathrm{total}}=\sum x^{2}-\frac{\left(\sum x\right)^{2}}{n}\)

\(S S_{\text {within}}=S S_{\text {total}}-S S_{\text {between}}\)

\(d f_{\mathrm{between}}=d f(n u m)=k-1\)

\(d f_{\text {within}}=d f(\text {denom})=n-k\)

\(M S_{\text {between}}=\frac{S S_{\text {between}}}{d f_{\text {between}}}\)

\(M S_{\text {within}}=\frac{S S_{\text {within}}}{d f_{\text {within}}}\)

\(F=\frac{M S_{\text {between}}}{M S_{\text {within}}}\)

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