For the theory section, show your work and do not round off your middle work. 

Keep your final answer to four (4) decimal places, where relevant.

1. Why is randomized block design preferred over completely randomized design?

2. You want to compare the abilities of four portfolio managers, A, B, C and D. One way to do so is to randomly allocate 40 stocks, ten to each of the portfolio managers. Each manager would then predict the value of the stocks in six months. You record their predicted values. After six months, you see what the values of the stocks are in reality, and determine the error in prediction.

a. Discuss the problems with using a completely randomized design for this experiment.

b. Explain how you could employ a randomized block design.

3. A food scientist is interested in comparing three recipes for banana bread. Four loaves of each recipe are to be baked in an oven that can bake three loaves at one time. As there may be variation in baking condition from one period to the other, a randomized block design was employed. The data are given below:

                Recipe

                A B C

Baking Period 1 51 49 44

(Minutes)   2 61 62 55

                  3 56 59 63

                        4 63 62 62

a. Write an appropriate statistical model for this experimental situation. 

b. Run an analysis of variance to compare the three recipes. Use  = 0.05. Summarize your results in an AOV table. 

c. Compute the relative efficiency of the randomized block design and the completely randomized design.

4. The data below were obtained for a randomized block design involving three treatments and five blocks:

        SSBlock = 50 SSTreatment = 60 SSError = 20. 

Set up an AOV table and test for any significant differences. 

5. Complete the assignments of treatments so that the design shown below is a Latin square design.

Column

A C B E D

Row B A C

E C B

E A C

A

Note: Copy the table onto your answer sheet, and write in the missing treatments in a different-coloured ink.

6. Write a short explanation of Latin square designs. Include an example of a 4 × 4 LSD. What are the advantages and disadvantages of LSD?

7. An experimenter used a 5 × 5 Latin square design for his study The results are shown below. 

Row 1 2 3 4 5 Total

1 A 20 D 35 B 27 C 21 E 23 126

2 D 21 C 16 E 16 A 23 B 32 108

3 B 20 E 10 A 18 D 30 C 25 103

4 C 20 B 30 D 24 E 16 A 31 121

5 E 23 A 25 C 19 B 22 D 36 125

Total 104      116      104      112      147 583

Treatment A B C D E

Totals 117 131 101 146   88

Means 23.4 26.2 20.2 29.2 17.6

a. If total sum of squares is 971.44, summarize your results in an AOV table.

b. Is there significant evidence that the treatment means are not equal?

c. Calculate the relative efficiency of the LSD to a CRD. 

8. An experiment was conducted to compare four different laundry detergents and four different washing times. The response variable was a measure of whiteness of pre-soiled clothes washed by each of the 16 treatment combinations. Each treatment combination was replicated three times, a total of 48 washings. The F ratios of for the main effects and interaction are shown below.

Effect F-ratio

Detergent 2.23

Time         3.65

Det × Time    3.16

Give the range of p-values within which the p-value for each of these F values falls. For each one, state the appropriate numerator and denominator degree of freedom. Comment on the result.