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
49.8	55.6	39.1
43.2	25.0	37.7
32.2	41.4	10.5
27.1	29.5	32.1
38.6	39.5	15.6
36.3		42.4
20.2

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 5% level of significance.

(a) What is the level of significance?

State the null and alternate hypotheses.

H_o: ?₁ = ?₂ = ?₃; H₁: All three means are different.

H_o: ?₁ = ?₂ = ?₃; H₁: At least two means are equal.     

H_o: ?₁ = ?₂ = ?₃; H₁: Exactly two means are equal.

H_o: ?₁ = ?₂ = ?₃; H₁: Not all the means are equal.

(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. (Use 4 decimal places.)

(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 ? = 0.05, we do not reject H₀.

Since the P value is less than or equal to the level of significance at ? = 0.05, we reject H₀.     

Since the P value is greater than the level of significance at ? = 0.05, we reject H₀.

Since the P value is less than or equal to the level of significance at ? = 0.05, we do not reject H₀.

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

At the 5% level of significance there is insufficient evidence to conclude that the means are not all equal.

At the 5% level of significance there is sufficient evidence to conclude that the means are all equal.     

At the 5% level of significance there is insufficient evidence to conclude that the means are all equal.

At the 5% 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						Do not reject H0/Reject H0.
Within groups
Total

Answer 1

a) LOS alpha = 0.05

b) Option (D) Ho: ?1 = ?2 = ?3; H1: Not all the means are equal.

c) From the given data

	1	2	3	4	5	6	7
								Total (Ti.)	Ti.^2/ni
I	49.8	43.2	32.2	27.1	38.6	36.3	20.2	247.4	8743.822857
II	55.6	25	41.4	29.5	39.5			191	7296.2
III	39.1	37.7	10.5	32.1	15.6	42.4		177.4	5245.126667
					G =			615.8	21285.14952

dfBET = 3-1 = 2

dfW = 15

MSBET = 217.9473/2 = 109

MSW = 2040.7704/15 = 136

Value of the sample F statistic = 109 / 136 = 0.80

df = (2,15)

P-value =  0.467

correct answer: option (A) Since the P value is greater than the level of significance at ? = 0.05, we do not reject H0.

Correct answer: At the 5% level of significance there is sufficient evidence to conclude that the means are all equal.