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Statistics LibreTexts

12.6: Chapter 12 Formula Review

  • Page ID
    6153
  • 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)