Question

#15

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	13.2	22.7	17.9
23.4	9.2	20.4	16.9
14.3	11.5	7.2	14.8
8.4	8.2	12.7	15.1
11.8	6.5	7.0	10.2
	19.8		9.8

(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)

(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					NA	NA
Within groups
Total

Answer 1

One Way Analysis of Variance (ANOVA)              

	I	II	III	IV
count, ni =	5	6	5	6
mean , x̅ i =	17.040	11.40	14.00	14.12
std. dev., si =	7.990	4.752	7.307	3.39
sample variances, si^2 =	63.833	22.580	53.395	11.494
total sum	85.2	68.4	70	84.7	308.3	(grand sum)
grand mean , x̅̅ =	Σni*x̅i/Σni =				14.01
square of deviation of sample mean from grand mean,( x̅ - x̅̅)²	9.159	6.831	0.000	0.011
					TOTAL
SS(between)= SSB = Σn( x̅ - x̅̅)² =	45.794	40.987	0.001	0.064	86.84558
SS(within ) = SSW = Σ(n-1)s² =	255.332	112.900	213.580	57.468	639.280

no. of treatment , k =   4
df between = k-1 =    3
N = Σn =   22
df within = N-k =   18
  
mean square between groups , MSB = SSB/k-1 =    28.9485
  
mean square within groups , MSW = SSW/N-k =    35.5156
  
F-stat = MSB/MSW =    0.8151
P value =   0.5022

---------------

SSTOT	=	726.126
SSBET	=	86.8
SSW	=	639.280

dfBET	=	3
dfW	=	18
MSBET	=	28.949
MSW	=	35.516

value of the sample F statistic=0.815

(numerator) DF=3
(denominator) DF = 18

ANOVA
Source of Variation	SS	df	MS	F	P-value	test decision
Between Groups	86.846	3	28.949	0.815	>0.10	do not reject Ho
Within Groups	639.280	18	35.516
Total	726.126	21