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 49.7 55.4 38.6 43.8 24.6 37.9 32.9 41.6 10.7 27.2 29.2 32.1 38.5 39.1 15.7 36.6 42.1 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? State the null and alternate hypotheses. Ho: μ1 = μ2 = μ3; H1: All three means are different. Ho: μ1 = μ2 = μ3; H1: Not all the means are equal. Ho: μ1 = μ2 = μ3; H1: Exactly two means are equal. Ho: μ1 = μ2 = μ3; H1: At least two means are equal. (b) Find SSTOT, SSBET, and SSW and check that SSTOT = SSBET + SSW. (Use 3 decimal places.) SSTOT = SSBET = SSW = Find d.f.BET, d.f.W, MSBET, and MSW. (Use 3 decimal places for MSBET, and MSW.) dfBET = dfW = MSBET = MSW = Find the value of the sample F statistic. (Use 3 decimal places.) What are the degrees of freedom? (numerator) (denominator)

Answer 1

I used minitab to solve this problem. First enter your data in minitab as follows:

Output is as follows:

One-way ANOVA: value versus factror

Method

Null hypothesis All means are equal
Alternative hypothesis At least one mean is different
Significance level α = 0.05

Equal variances were assumed for the analysis.

Factor Information

Factor Levels Values
factror 3 1, 2, 3

Analysis of Variance

Source DF Adj SS Adj MS F-Value P-Value
factror 2 521.7 260.8 2.30 0.135
Error 15 1701.8 113.5
Total 17 2223.5

Model Summary

S R-sq R-sq(adj) R-sq(pred)
10.6515 23.46% 13.26% 0.00%

Means

factror N Mean StDev 95% CI
1 6 41.60 10.60 (32.33, 50.87)
2 6 28.95 10.22 (19.68, 38.22)
3 6 32.05 11.12 (22.78, 41.32)

Pooled StDev = 10.6515

a)

Level of significance is the probability of rejecting null hypothesis when it is true.

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

b)

SSTOT= 2223.5

SSBET= 521.7   

SSW= 1701.8

SSTOT = SSBET + SSW relation hold

dfbet =  2 d.f.w= 15

MSBET= 260.8     MSW= 113.5

F stat= MSBET/ MSW =  2.30

Numerator df= 2 denominator df=15

  

b)