Question

How productive are U.S. workers? One way to answer this question is to study annual profits per employee. A random sample of companies in computers (I), aerospace (II), heavy equipment (III), and broadcasting (IV) gave the following data regarding annual profits per employee (units in thousands of dollars).

I	II	III	IV
27.3	13.6	22.9	17.2
23.8	9.1	20.6	16.1
14.1	11.3	7.3	14.3
8.9	8.6	12.4	15.5
11.9	6.6	7.6	10.5
	19.7		9.9

Shall we reject or not reject the claim that there is no difference in population mean annual profits per employee in each of the four types of companies? Use a 5% level of significance.

(a) What is the level of significance?

State the null and alternate hypotheses (from the following).

H_o: μ₁ = μ₂ = μ₃ = μ₄; H₁: Not all the means are equal.

H_o: μ₁ = μ₂ = μ₃ = μ₄; H₁: Exactly three means are equal.    

H_o: μ₁ = μ₂ = μ₃ = μ₄; H₁: Exactly two means are equal.

H_o: μ₁ = μ₂ = μ₃ = μ₄; H₁: All four 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.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis?
(e) Interpret your conclusion in the context of the application.

(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 from the following--- 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 from the following--- Do not reject H0. -- Reject H0.
Within groups
Total

Answer 1

a)

level of significance =0.05

H_o: μ₁ = μ₂ = μ₃ = μ₄; H₁: Not all the means are equal.

b)

SSTOT	708.395
SSBET	89.306
SSW	619.089

dfBET	3
dfW	18
MSBET	29.769
MSW	34.394

value of test statistic for factor A =

0.87

df(numerator) =		3
df(Denominator) =		18

c)

p value =0.4770

d) r fail to reject the null hypothesis

e)can not conclude that population means are not equal

f)

Source	SS	df	MS	F	P value
Between	89.306	3	29.769	0.866	0.4770
Within	619.089	18	34.394
Total	708.395	21