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
49.3	55.7	38.7
43.3	25.0	37.3
32.6	41.5	10.9
27.1	29.4	32.9
38.5	39.9	15.5
36.5		42.2
20.4

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?


State the null and alternate hypotheses.

H_o: μ₁ = μ₂ = μ₃; H₁: Not all the means are equal.H_o: μ₁ = μ₂ = μ₃; H₁: Exactly two means are equal.    H_o: μ₁ = μ₂ = μ₃; H₁: At least two means are equal.H_o: μ₁ = μ₂ = μ₃; H₁: All three means are different.

(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) Find the P-value of the sample test statistic.

P-value > 0.1000.050 < P-value < 0.100    0.025 < P-value < 0.0500.010 < P-value < 0.0250.001 < P-value < 0.010P-value < 0.001

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis?

Since the P-value is greater than the level of significance at α = 0.01, we do not reject H₀.Since the P-value is less than or equal to the level of significance at α = 0.01, we reject H₀.    Since the P-value is greater than the level of significance at α = 0.01, we reject H₀.Since the P-value is less than or equal to the level of significance at α = 0.01, we do not reject H₀.

(e) Interpret your conclusion in the context of the application.

At the 1% level of significance there is insufficient evidence to conclude that the means are not all equal.At the 1% level of significance there is sufficient evidence to conclude that the means are all equal.    At the 1% level of significance there is insufficient evidence to conclude that the means are all equal.At the 1% level of significance there is sufficient evidence to conclude that the means are not all equal.

(f) 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					---Select--- p-value > 0.100 0.050 < p-value < 0.100 0.025 < p-value < 0.050 0.010 < p-value < 0.025 0.001 < p-value < 0.010 p-value < 0.001	---Select--- Do not reject H0. Reject H0.
Within groups
Total

Answer 1

Applying one way ANOVA: (use excel: data: data analysis: one way ANOVA: select Array):

a)

level of significance =0.01

H_o: μ₁ = μ₂ = μ₃; H₁: Not all the means are equal.

b)

SSTOT	2221.783
SSBET	221.706
SSW	2000.077

dfBET	2
dfW	15
MSBET	110.853
MSW	133.338

value of test statistic for factor A =

0.831

df(numerator) =		2
df(Denominator) =		15

c)

P-value > 0.100

d)

Since the P-value is greater than the level of significance at α = 0.01, we do not reject H₀
e)

At the 1% level of significance there is insufficient evidence to conclude that the means are not all equal

f)

Source	SS	df	MS	F	P value
treatment	221.706	2	110.853	0.831	0.4546
error	2000.077	15	133.338
Total	2221.783	17