Question

A random sample of companies in electric utilities (I), financial services (II), and food processing (III) gave the following information regarding annual profits per employee (units in thousands of dollars).

I	II	III
49.4	55.4	38.8
43.4	25.2	37.8
32.1	41.9	10.4
27.3	29.9	32.5
38.7	39.6	15.7
36.3		42.3
20.3

Shall we reject or not reject the claim that there is no difference in population mean annual profits per employee in each of the three types of companies? Use a 1% level of significance.

(a). what is the level of significance

(b).  Find SS_TOT, SS_BET, and SS_W and check that SS_TOT = SS_BET + SS_W. (Use 3 decimal places.)

SSTOT	=
SSBET	=
SSW	=

Find d.f._BET, d.f._W, MS_BET, and MS_W. (Use 3 decimal places for MS_BET, and MS_W.)

dfBET	=
dfW	=
MSBET	=
MSW	=

Find the value of the sample F statistic. (Use 3 decimal places.)

What are the degrees of freedom?
(numerator)
(denominator)

(c). Make a summary table for your ANOVA test.

Source of Variation	Sum of Squares	Degrees of Freedom	MS	F Ratio	P Value	Test Decision
Between groups
Within groups
Total

Answer 1

Consider

y₁: Annual profits per employee in electrical utilities

y₂: Annual profits per employee in financial services.

y₃: Annual profits per employee in food processing company.

We have to test the hypothesis

There is no significant difference in population mean annual profits per employee in each of the three types of company at 1% level of significance.

i.e. Null Hypothesis -

against

Alternative hypothesis - Ha: At least two are different.

a) Alpha = level of significance = 0.01.

b )

Y1	Y2	Y3	(Y1-Y1bar)^2	(Y2-Y2bar)^2	(Y3-Y3bar)^2
49.4	55.4	38.8	197.2030	289	84.9476
43.4	25.2	37.8	64.6882	174.24	67.5142
32.1	41.9	10.4	10.6087	12.25	367.9990
27.3	29.9	32.5	64.9169	72.25	8.5071
38.7	39.6	15.7	11.1750	1.44	192.7460
36.3		42.3	0.8891		161.7145
20.3			226.7163
247.5	192	177.5	576.1971	549.1800	883.4283

n₁ = 7 , n2= 5 , n₃ = 6

N = 7 + 5+ 6= 18

G = Grand Total = 247.5 + 192 + 177.5 = 617

Ybar = Grand Mean = 617/18 = 34.2778

y₁bar = 247.5 / 7 = 35.3571.

y₂bar =192 / 5 = 38.40.

y₁bar = 177.5 / 6= 29.5833.

Correction factor =C.F= G² / N = 617² / 18 = 21149.3889.

Raw Sum of square (RSS) = 49.4 ² + .............. + 42.3 ² = 23383.54

Total Sum of square = SSTOT = RSS - C.F. = 23383.54 - 21149.3889 = 2234.1511.

SSTOT = 2234.151.

Between Sum of square = SSBET

SSBET = n₁ * ( y₁bar-ybar)² + n₂ *(y₂bar-ybar)² + n₃ * ( y₃bar-ybar)².

= 7 * ( 35.3571 - 34.2778)²+ 5 * ( 38.40- 34.2778)²+ 6 * ( 29.5833 - 34.2778)²

SSBET = 225.346.

Within Sum of square = SSW

= Sum ( Y1 - Y1bar)² + Sum ( Y2-Y2bar)² + sum(Y3-Y3bar)²

= 576.1971 + 549.1800 + 883.4283

= 2008.805

SSW = 2008.805

SSBET +SSW = 225.346 + 2008.805 = 2234.151 = SSTOT.

Degrees of Freedom

Number of companies = k =3

dfBet = k-1 = 2

Total number of observations =N = 18.

dfW = N-K = 18 -3 = 15

MSBET = SSBET /dfBet = 225.346 / 2 = 112.673.

MSW =SSW/dfW = 2008.805 / 15 = 133.920.

Value of the sample F statistic is

F = MSBet / MSW ~ F _{k-1, N-k}.

F = 112.973 / 133.920 = 0.8413

Degrees of freedom for numerator = 2

Degrees of freedom for denominator = 15.

Since value of F-statistic is 0.8413

p-value is given by

p-value = P ( F2,15 > 0.8413) = 0.4505

Decision : Since p-value > level of significance, we fail to reject Ho .

Conclusion : There is sufficient evidence support to claim that there is no significant difference in population mean annual profits per employee in each of the three types of company.

c) Anova table:

Source of Variation	SS	df	MS	F	P-value	Test
Between Groups	225.3456	2	112.6728	0.841342	0.4505	Accept Ho.
Within Groups	2008.805	15	133.9204
Total	2234.151	17

Alternative Solution by using Excel

Y1	Y2	Y3
49.4	55.4	38.8
43.4	25.2	37.8
32.1	41.9	10.4		Anova: Single Factor
27.3	29.9	32.5
38.7	39.6	15.7		SUMMARY
36.3		42.3		Groups	Count	Sum	Average	Variance
20.3				Column 1	7	247.5	35.35714	96.03286
				Column 2	5	192	38.4	137.295
				Column 3	6	177.5	29.58333	176.6857
				ANOVA
				Source of Variation	SS	df	MS	F	P-value	F crit
				Between Groups	225.3456	2	112.6728	0.841342	0.450495	3.68232
				Within Groups	2008.805	15	133.9204
				Total	2234.151	17