Question

A marketing organization wishes to study the effects of four sales methods on weekly sales of a product. The organization employs a randomized block design in which three salesman use each sales method. The results obtained are given in the following table, along with the Excel output of a randomized block ANOVA of these data.

	Salesman, j
Sales Method, i	A	B	C
1	38	29	28
2	38	32	28
3	33	23	16
4	32	20	14

ANOVA: Two-Factor without Replication
SUMMARY	Count	Sum	Average	Variance
Method 1	3	95	31.6667	30.3333
Method 2	3	98	32.6667	25.3333
Method 3	3	72	24.0000	73.0000
Method 4	3	66	22.0000	84.0000
Salesman A	4	141	35.25	10.2500
Salesman B	4	104	26.00	30.0000
Salesman C	4	86	21.50	57.0000

ANOVA
Source of Variation	SS	df	MS	F	P-Value	F crit
Rows	259.5833	3	86.5278	16.14	.0028	4.7571
Columns	393.1667	2	196.5833	36.67	.0004	5.1433
Error	32.1667	6	5.36111
Total	684.9167	11

(a) Test the null hypothesis H₀ that no differences exist between the effects of the sales methods (treatments) on mean weekly sales. Set α = .05. Can we conclude that the different sales methods have different effects on mean weekly sales?

F = 16.14, p-value = .0028; (Do not reject OR Reject) H₀: there is (Difference OR No difference )a in effects of the sales methods (treatments) on mean weekly sales.

(b) Test the null hypothesis H₀ that no differences exist between the effects of the salesmen (blocks) on mean weekly sales. Set α = .05. Can we conclude that the different salesmen have different effects on mean weekly sales?

F = 36.67, p-value = .0004; ((Do not reject OR Reject) t H₀: salesman (Do not OR Do)do notdo have an effect on sales

(c) Use Tukey simultaneous 95 percent confidence intervals to make pairwise comparisons of the sales method effects on mean weekly sales. Which sales method(s) maximize mean weekly sales? (Round your answers to 2 decimal places. Negative amounts should be indicated by a minus sign.)

Method 1 – Method 2:	[, ]
Method 1 – Method 3:	[, ]
Method 1 – Method 4:	[, ]
Method 2 – Method 3:	[, ]
Method 2 – Method 4:	[, ]
Method 3 – Method 4:	[, ]

Answer 1

a)

p-value = .0028<α=0.05

; (Do not reject OR Reject) H₀: there is (Difference ) in effects of the sales methods (treatments) on mean weekly sales.

b)

p-value = .0004<α=0.05;

(Reject) H₀: salesman do have an effect on sales

c)

	method 1	method 2	method 3	method 4
count, ni =	3	3	3	3
mean , x̅ i =Σxi / ni	31.667	32.667	24.000	22.000

Level of significance	0.05
no of treatments	4
df error	6
MSE	5.3611
q-statistic value	4.898

Tukey Kramer test
critical value = q*√(MSE/2*(1/ni+1/nj))

confidence interval = mean difference ± critical value

if confidence interval contans zero, then means are not different.

			confidence interval
population mean difference		critical value	lower limit	upper limit	result
µ1-µ2	-1.00	6.55	-7.55	5.55	means are not different
µ1-µ3	7.67	6.55	1.12	14.21	means are different
µ1-µ4	9.67	6.55	3.12	16.21	means are different
µ2-µ3	8.67	6.55	2.12	15.21	means are different
µ2-µ4	10.67	6.55	4.12	17.21	means are different
µ3-µ4	2.00	6.55	-4.55	8.55	means are not different