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12.5：章节公式回顾

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    204468

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    12.1 两个方差的检验

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

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

    如果\(\delta_{0}=1\)那么

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

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

    测试统计数据为：

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

    12.3 F 分布和 F 比率

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

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

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

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

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

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

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

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

    • \(k\)= 组数
    • \(n_j\)= 第 j 组的大小
    • \(s_j\)= 第 j 组中值的总和
    • \(n\)= 所有值（观测值）的总数
    • \(x\)= 来自数据的一个值（一个观测值）
    • \(s_{\overline{x}}^{2}\)= 样本均值的方差
    • \(s^2_{pooled}\)= 样本方差的平均值（合并方差）