A video game developer is testing a new game on three different groups. Each group represents a different target market for the game. The developer collects scores from a random sample from each group. The results are shown in the Table.

$\begin{array}{|lll|}\hline \text{Group A}& \text{Group B}& \text{Group C}\\ 101& 151& 101\\ 108& 149& 109\\ 98& 160& 198\\ 107& 112& 186\\ 111& 126& 160\\ \hline\end{array}$

What is$p$-value?

From our Table, we can calculate

$s_1=525,s_2=698,s_3=754$

We now can calculate:

${\mathbf{SS}}_{\text{between}}=\sum \left[\frac{{\left(s_j\right)}^2}{n_j}\right]-\frac{{\left(\sum s_j\right)}^2}{n}$

$S{S}_{\text{between}}=\frac{s_1^2}{5}+\frac{s_2^2}{5}+\frac{s_3^2}{5}-\frac{{\left(s_1+s_2+s_3\right)}^2}{15}$

where $n_1=n_2=n_3=5$and $n=n_1+n_2+n_3=15$

So, we calculate

$\frac{(525{)}^2}{5}+\frac{(698{)}^2}{5}+\frac{(754{)}^2}{5}-\frac{(525+698+754{)}^2}{15}$

${\mathbf{SS}}_{\text{between}}=266.269-260568.6=\mathbf{5700}\mathbf{.}\mathbf{4}$

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

$S_{\text{total}}=275.743-260568.6$

${\mathbf{S}}_{\text{total}}=\mathbf{15174}\mathbf{.}\mathbf{4}$

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

${\mathbf{SS}}_{\text{within}}=\mathbf{9474}$

$M{S}_{\text{between}}=\frac{S{S}_{\text{between}}}{k-1}$, where $k=3$

${\mathbf{MS}}_{\text{between}}=\frac{5700.4}{3}=\mathbf{2850}\mathbf{.}\mathbf{2}$

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

${\mathbf{MS}}_{\text{within}}=\frac{9474}{12}=\mathbf{789}\mathbf{.}\mathbf{5}$

Now, we can find F-statistic:

$\mathrm{F}=\frac{M{S}_{\text{between}}}{M{S}_{\text{within}}}=\frac{2850.2}{789.5}$

$F=3.61$

To find these results on the calculator:

Press START. Press $1$EDIT. Put the data into the lists $L_1,L_2,L_3$

Press START, and arrow over to TESTS, and arrow down to ANOVA. Press ENTER, and then enter $L_1,L_2,L_3$. Press ENTER. We will see that the values in the foregoing ANOVA table are easily produced by the calculator, including the test statistic and the $p$-value of the test.

The calculator displays:

$F=3.610132996$

$\mathbf{p}=\mathbf{0}\mathbf{.}\mathbf{05922855}(p-value)$

Factor

$df=2$

$SS=5700.4$

$MS=2850.2$

Error

$df=12$

$SS=9474$

$MS=789.5$