 Solution

The reduced model combines the 0-, 3-, and 6-point categories. Relative to the full model, the null hypothesis is ${\eta }_3={\eta }_6=0$, leaving the 9-point contrast unrestricted. These are two linear restrictions on an otherwise unchanged normal linear model.

For nested normal models, the reduction in residual sum of squares divided by ${\sigma }^2$ has a chi-squared distribution with degrees of freedom equal to the number of restrictions. It is independent of the full model's residual variance estimate. Hence the F-test statistic isIt is below the supplied 5% critical value $3.402826$, so do not reject equal effects for 0, 3, and 6 points. There is insufficient evidence that separating those categories improves this additive model; this does not prove that their true premiums are identical.