Question

Topic: ANOVA

Topic: ANOVA

1- An experiment was conducted using a randomized block design. The data from the experiment are displayed in the following table.

	Block
Treatment	1	2	3
1	2	3	5
2	8	6	7
3	7	6	5

a) Fill in the missing entries in the ANOVA table.

Source	df	SS	MS	F
Treatment	2	21.5555
Block	2
Error	4
Total	8	30.2222

b) Specify the null use to investigate whether a difference exists among the treatment means.

c) What test statistic should be used in conducting the test of part b?

d) Describe the Type I and Type II errors associated with the hypothesis test of part (5 marks)

e) Conduct the hypothesis test of part b using alpha = .05. (5 marks)

2- Suppose an experiment employing a randomized block design has four treatments and nine blocks, for a total of 4 * 9 = 36 observations. Assume that the Total Sum of Squares for the response is SS(Total) = 500. For each of the following partitions of SS(Total), test the null hypothesis that the treatment means are equal and test the null hypothesis that the block means are equal. Use alpha = .05 for each test.

a) The Sum of Squares for Treatments (SST) is 20% of SS(Total), and the Sum of Squares for Blocks (SSB) is 30% of SS(Total). (2 marks)

b) SST is 50% of SS(total), and SSB is 20% of SS(total)

c) SST is 20% of SS(total), and SSB is 50% of SS(Total)

d) SST is 40% of SS(total), and SSB is 40% of SS(Total)

e) SST is 20% of SS(total), and SSB is 20% of SS(Total)

Answer 1

Question 1

(b)

ANOVA
Source of Variation	SS	df	MS	F
Treatment	21.55556	2	10.77778	5.542857
Block	0.888889	2	0.444444	0.228571
Error	7.777778	4	1.944444
Total	30.22222	8

Here block error will be calculated by SS(BLock) = $\sum n(x_{GM} -\bar x)^2$

= 3 * [(5.667 - 5.4444)² + (5 - 5.4444)² + (5.6667 - 5.4444)²] = 0.888889

so SS(ERROR) = SS(Total) - SS(Block) - SS(Treatment) = 7.77778

(c) Here F test statistic is used.

(d) Here type I error is when we reject the null hypothesis that population means are different when it is not. Type II error here is that we fail to reject the null hypothesis that population means are same when they are not the same.

(e) Here F(critical) for dF_block = df_treatment = 2 and dF (error) = 4

so here F(critical) = 6.9444

so here F < F(critical) for the treatment and the block terms so we fail to reject the null hypothesis for treatment and error also.