What is the $F-$statistic?

Four basketball teams took a random sample of players regarding how high each player can jump (in inches).

To find the F- statistic:

From our Table, we can calculate

$s_1=129,s_2=105,s_3=137,s_4=128,s_5=120$

We now can calculate:

localid="1649062798912" ${\mathbf{SS}}_{\text{Factors}}=\sum \left[\frac{{\left(s_j\right)}^2}{n_j}\right]-\frac{{\left(\sum s_j\right)}^2}{n}$$S_{\text{Factors}}=\frac{s_1^2}{3}+\frac{s_2^2}{3}+\frac{s_3^2}{3}+\frac{s_4^2}{3}+\frac{s_5^2}{3}-\frac{{\left(s_1+s_2+s_3+s_4+s_5\right)}^2}{15}$

localid="1649062832564" $\text{where}{n}_1=n_2=n_3=n_4=n_5=3\text{and}n=n_1+n_2+n_3+n_4+n_5=15$

So, we calculate

localid="1649062843677" $\frac{(129{)}^2}{3}+\frac{(105{)}^2}{3}+\frac{(137{)}^2}{3}+\frac{(128{)}^2}{3}+\frac{(120{)}^2}{3}-\frac{(129+105+137+128+120{)}^2}{15}$

${\mathbf{SS}}_{\text{Factors}}=25739.6667-25544.0667=\mathbf{195}\mathbf{.}\mathbf{6}$

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

$S_{\text{total}}=25977-25544.0667$

$S_{\text{total}}=432.93334$

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

${\mathbf{SS}}_{\text{within}}\mathbf{=}\mathbf{237}\mathbf{.}\mathbf{3334}$

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

$M{S}_{\text{Factors}}=\frac{S{S}_{\text{Factors}}}{k-1},\text{where}\mathrm{k}=5$

${\mathbf{MS}}_{\text{Factors}}=\frac{195.6}{4}=\mathbf{48}\mathbf{.}\mathbf{9}$

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

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

${\mathbf{MS}}_{\text{within}}=\frac{237.3334}{10}=\mathbf{23}\mathbf{.}\mathbf{73334}$

Now, we can find F-statistic:

$\mathrm{F}=\frac{M{S}_{\text{Factor}}}{M{S}_{\text{within}}}=\frac{48.9}{23.73334}$

$=2.06$