Question

What is the $F$ statistic?

Short answer

The value of$F-statistic=10.404$.

Step-by-step solution

Given Information

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

Calculate the F statistics

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,

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

Calculate the sum of squares within the group

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

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

$S_{\text{total}}=123-84.05$

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

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

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

Calculate the mean of square

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:

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

$F=10.404$