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Q.22

Expert-verifiedFound in: Page 766

Book edition
OER 2018

Author(s)
Barbara Illowsky, Susan Dean

Pages
902 pages

ISBN
9781938168208

What is the $F$ statistic?

The value of $F-statistic=10.404$.

Team 1 | Team 2 | Team 3 | Team 4 |

1 | 2 | 0 | 3 |

2 | 3 | 0 | 4 |

0 | 2 | 1 | 4 |

3 | 4 | 1 | 3 |

2 | 4 | 0 | 2 |

To find the value of $\text{F}$-statistic:

From our Table, we can calculate

${s}_{1}=8,{s}_{2}=15,{s}_{3}=2,{s}_{4}=16$We now can calculate:

${\mathbf{SS}}_{\text{between}}=\sum \left[\frac{{\left({s}_{j}\right)}^{2}}{{n}_{j}}\right]-\frac{{\left(\sum {s}_{j}\right)}^{2}}{n}$$S{S}_{\text{between}}=\frac{{s}_{1}^{2}}{5}+\frac{{s}_{2}^{2}}{5}+\frac{{s}_{3}^{2}}{5}+\frac{{s}_{4}^{2}}{5}-\frac{{\left({s}_{1}+{s}_{2}+{s}_{3}+{s}_{4}\right)}^{2}}{20}$

$\text{where}{n}_{1}={n}_{2}={n}_{3}={n}_{4}=5\text{and}n={n}_{1}+{n}_{2}+{n}_{3}+{n}_{4}=20$

We calculate,

$\frac{(8{)}^{2}}{5}+\frac{(15{)}^{2}}{5}+\frac{(2{)}^{2}}{5}+\frac{(16{)}^{2}}{5}-\frac{(8+15+2+16{)}^{2}}{20}$

${\mathbf{SS}}_{\text{between}}=109,8-84.05=\mathbf{25}\mathbf{.}\mathbf{75}$

${\mathbf{S}}_{\text{total}}=\sum {x}^{2}-\frac{{\left(\sum x\right)}^{2}}{n}$

${S}_{\text{total}}=123-84.05$

${\mathbf{S}}_{\text{total}}=\mathbf{38}\mathbf{.}\mathbf{95}$

${\mathbf{SS}}_{\text{within}}={\mathbf{SS}}_{\text{total}}-{\mathbf{SS}}_{\text{between}}$

${\mathrm{SS}}_{\text{within}}=13.2$

We calculate $M{S}_{\text{between}}$ like:

$M{S}_{\text{between}}=\frac{S{S}_{\text{between}}}{k-1},\text{where}\mathrm{k}=4$

${\mathbf{MS}}_{\text{between}}=\frac{25.75}{3}=\mathbf{8}\mathbf{.}\mathbf{58333}$

We calculate $M{S}_{\text{within}}$ like:

$M{S}_{\text{within}}=\frac{S{S}_{\text{within}}}{n-k}$

${\mathbf{MS}}_{\text{within}}=\frac{13.2}{16}=\mathbf{0}\mathbf{.}\mathbf{825}$

Now, we can find $F-$statistic:

$\mathrm{F}=\frac{M{S}_{\text{between}}}{M{S}_{\text{within}}}=\frac{8.58333}{0.825}$$F=10.404$

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