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	39	31	23
2	41	28	28
3	31	23	15
4	33	19	16

ANOVA: Two-Factor without Replication
SUMMARY	Count	Sum	Average	Variance
Method 1	3	93	31.0000	64.0000
Method 2	3	97	32.3333	56.3333
Method 3	3	69	23.0000	64.0000
Method 4	3	68	22.6667	82.3333
Salesman A	4	144	36.00	22.6667
Salesman B	4	101	25.25	28.2500
Salesman C	4	82	20.50	37.6667

ANOVA
Source of Variation	SS	df	MS	F	P-Value	F crit
Rows	236.9167	3	78.9722	16.43	.0027	4.7571
Columns	504.5000	2	252.2500	52.49	.0002	5.1433
Error	28.8333	6	4.80556
Total	770.2500	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 = 16.43, p-value = .0027; (Click to select)RejectDo not reject H₀: there is (Click to select)a differenceno difference 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 = 52.49, p-value = .0002; (Click to select)Do not rejectReject H₀: salesman (Click to select)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) 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 = 16.43, p-value = .0027; (Click to select)Reject H₀: there is (Click to select)a difference 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 = 52.49, p-value = .0002; (Click to select)Reject H₀: salesman (Click to select)do 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:	[-7.53,4.87 ]
Method 1 – Method 3:	[1.80, 14.20]
Method 1 – Method 4:	[2.13,14.53 ]
Method 2 – Method 3:	[3.13, 15.53]
Method 2 – Method 4:	[3.47, 15.87]
Method 3 – Method 4:	[-5.87,6.53 ]

MINITAB used:

General Linear Model: sales versus Salesman, Method

Method

Factor coding

(-1, 0, +1)

Factor Information

Factor	Type	Levels	Values
Salesman	Fixed	3	A, B, C
Method	Fixed	4	1, 2, 3, 4

Analysis of Variance

Source	DF	Adj SS	Adj MS	F-Value	P-Value
Salesman	2	504.50	252.250	52.49	0.0002
Method	3	236.92	78.972	16.43	0.0027
Error	6	28.83	4.806
Total	11	770.25

Model Summary

S	R-sq	R-sq(adj)	R-sq(pred)
2.19216	96.26%	93.14%	85.03%

Comparisons for sales

Tukey Pairwise Comparisons: Method

Grouping Information Using the Tukey Method and 95% Confidence

Method	N	Mean	Grouping
2	3	32.3333	A
1	3	31.0000	A
3	3	23.0000		B
4	3	22.6667		B

Means that do not share a letter are significantly different.

Tukey Simultaneous Tests for Differences of Means

Difference of Method Levels	Difference of Means	SE of Difference	Simultaneous 95% CI	T-Value	Adjusted P-Value
2 - 1	1.33	1.79	(-4.87, 7.53)	0.74	0.876
3 - 1	-8.00	1.79	(-14.20, -1.80)	-4.47	0.017
4 - 1	-8.33	1.79	(-14.53, -2.13)	-4.66	0.014
3 - 2	-9.33	1.79	(-15.53, -3.13)	-5.21	0.008
4 - 2	-9.67	1.79	(-15.87, -3.47)	-5.40	0.007
4 - 3	-0.33	1.79	(-6.53, 5.87)	-0.19	0.997

Individual confidence level = 98.66%