# ANOVA table and hypothesis testing The interest is in the (differences between) means, which we want to test using the null hypothesis of no difference between means:

H_0:μ_1=μ_2=…=μ_t with the alternative ?(H?_a) of at least one being different.

To test H_0 we use the ratio F=MSTreat/MSE

MSTreat: Mean Squares Treatments, MSTreat=SSTreat/(t-1)

Under the H_0 MSTreat is an estimator of σ^2 but it is larger than σ^2 when there is a difference between treatment means).

MSE: Mean Squares of the Error, MSE=SSE/(N-t)

MSE is another estimator of σ^2.

Under the null hypothesis (no difference between means) both MSTreat and MSE are estimators of σ^2 and its ratio follows an F distribution: F~F_(t-1,N-t)

Large values of F support H_a. We use p-value from the computer output (or tables to find critical region).

An equivalent expression of the null hypothesis can be given in terms of treatment effects: H_0:τ_1=τ_2=…=τ_t=0, and alternative ?(H?_a) of at least one being different from zero.