Question

A Businessman interested in testing to see a difference in the average deals processed by the 3 adopted systems

a. report the descriptive statistics for each sample

b. report and analyze the ANOVA test for the 3 systems

Date	Birmingham	Memphis	Little Rock	Jackson
January 9	18	20	18	14
January 17	23	31	22	30
January 18	19	25	22	21
January 31	29	36	28	35
February 1	27	31	28	24
February 6	26	31	31	25
February 14	31	24	19	25
February 17	31	31	28	28
February 20	33	35	35	34
February 29	20	42	42	21

Answer 1

the descriptive statistics for each sample

Factor	N	Mean $\bar{X}_{j}$	StDev $s_{j}$	95%	CI
Birmingham	10	25.7	5.44	(21.56,	29.84)
Memphis	10	30.6	6.38	(26.46,	34.74)
Little Rock	10	27.3	7.47	(23.16,	31.44)
Jackson	10	25.7	6.36	(21.56,	29.84)

In this example, the hypotheses are:

• H₀: μ₁ = μ₂ = μ₃ = μ₄
• H₁: The means are not all equal.

Null hypothesis There is no significant difference in the the average deals processed by the 3 adopted systems among the four cities.
Alternative hypothesis At least one average deals processed by the 3 adopted systems is different among the four cities.
Significance level α = 0.05

Equal variances were assumed for the analysis.

Factor Information

Factor Levels Values
City 4 Birmingham AL, Memphis TN, Little Rock AR, Jackson MS

The test statistic for testing H₀: μ₁ = μ₂ = ... = μ4 is:

$F=\frac{\frac{\sum_{j=1}^{4}n_{j}(\bar{X}_{j}-\bar{X})^{2}}{3}}{\frac{\sum_{i=1}^{4}\sum_{j=1}{4}(X_{ij}-\bar{X}_{j})^{2}}{36}}=1.28$

where

• 
  27
  = individual observation,

• 
  X
  = sample mean of the j^th treatment (or group),
• $equation_image45.gif$ = overall sample mean,
• k = the number of treatments or independent comparison groups, and
• N = total number of observations or total sample size.

In order to determine the critical value of F we need degrees of freedom, df₁=k-1 and df₂=N-k. In this example, df₁=k-1=4-1=3 and df₂=N-k=40-4=36.

The critical value is

F0.05,3,36 = 2.8662

and the decision rule is as follows: Reject H₀ if F > 2.8662.

Analysis of Variance

Source	DF	Adj SS	Adj MS	F-Value	P-Value
Factor	3	160.1	53.36	1.28	0.295
Error	36	1498.7	41.63
Total	39	1658.8

Conclusion:

We fail to reject H₀ because F-value=1.28 < F-critical value= 2.8662 and also P-value=0.295> level of significance=0.05.

We have statistically no significant evidence at α=0.05

Hence, There is no significant difference in the the average deals processed by the 3 adopted systems among the four cities.