Question

An experiment has been conducted for four treatments with eight blocks. Complete the following analysis of variance table (to 2 decimals, if necessary and p-value to 4 decimals). If your answer is zero enter "0".

Source of Variation	Sum of Squares	Degrees of Freedom	Mean Square	F	p-value
Treatments	1,100
Blocks	500
Error
Total	2,200

Use  = .05 to test for any significant differences

Answer 1

Let the number of treatments, and the number of blocks,

Total Sum of Squares, , where

SSTr - Sum of squares due to treatments = 1100

SSB - Sum of squares due to blocks, and = 500

SSE - Sum of squares due to error

Therefore,

Degrees of freedom for treatments = v-1 = 3

Degrees of freedom for blocks = b-1 = 7

Total degrees of freedom = n-1=vb-1=(4*8)-1=31

Degrees of freedom for error = 31 - (3+7) = 21

Mean sum of squares, MS = Sum of squares / degrees of freedom.

Source of Variation	Sum of Squares	Degrees of Freedom	Mean square	F-ratio	p-value
Treatments	1100	3	MSTr=366.667		5.5523e-05
Blocks	500	7	MSB=71.428		0.0491
Error	600	21	MSE=28.5714	---------------
Total	2200	31	----------	---------------

The R-codes to get the p-value

> 1-pf(12.833,3,21)
[1] 5.552365e-05
> 1-pf(2.4999,7,21)
[1] 0.04910295

We test for two hypothesis :

The first one is :

where is the ith treatment effect.

At 0.05 level the critical value to test for equality of treatment effects is :

which implies that we reject the null hypothesis and conclude that there is at least one pair of treatment effects that differ significantly.

The second one is :

where is the jth block effect.

At 0.05 level the critical value to test for equality of block effects is :

which implies that we reject the null hypothesis and conclude that there is at least one pair of block effects that differ significantly.