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.