ANOVA table insecticide example

Note that in the ANOVA table GenStat calls the within treatment source of variation as residual, which emphasises the fact that these are estimates of the error variation.

MSE = SSE / dferror = 1.990 / 21 = 0.095

MSE is an estimate of s2

MSTreat = SSTreat / dfTreat = 5.587 / 6 = 0.931

Use the ratio MSTreat and MSE for testing.

F (or v.r. for variance ratio in GenStat) = 0.931 / 0.095 = 9.83

To conclude we need to determine whether the observed F value is large or not. We use the probability that is printed in the output to decide (F pr.). The F pr. is the probability to observe an F value as large or larger than the one now observed under the H_0. When that value is too small based on some arbitrary level of significance, say 0.05, we conclude that H_0 is very unlikely to be true, and so we reject  H_0.

The observed value of the F=9.83 and the probability is <0.001, so we reject H_0 and conclude that there is at least one difference between the group means.

We have proven that at least two means are different.