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
27.5	13.8	22.1	17.6
23.1	9.5	20.3	16.8
14.3	11.3	7.7	14.3
8.9	8.3	12.5	15.1
11.1	6.1	7.9	10.6
	19.5		9.8

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.

H_o: μ₁ = μ₂ = μ₃ = μ₄; H₁: Not all the 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.H_o: μ₁ = μ₂ = μ₃ = μ₄; H₁: Exactly three 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.

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 = 0.05, we do not reject H₀.Since the P-value is less than or equal to the level of significance at α = 0.05, we reject H₀.    Since the P-value is greater than the level of significance at α = 0.05, we reject H₀.Since the P-value is less than or equal to the level of significance at α = 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					---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 ANOVA:

a)

level of significance =0.05

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

b)

SSTOT	688.050
SSBET	84.500
SSW	603.550

dfBET	3
dfW	18
MSBET	28.167
MSW	33.531

value of the sample F statistic =0.840

(numerator df =3

denominator df =18

c)

P-value > 0.100

d)

Since the P-value is greater than the level of significance at α = 0.05, we do not reject H₀.

e)

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

f)

Source of Variation	SS	df	MS	F	P-value
Between Groups	84.500	3	28.167	0.840	p value >0.10	do not reject HO
Within Groups	603.550	18	33.531
Total	688.050	21