# 12.5: Chapter 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)

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