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	13.7	22.8	17.1
23.3	9.3	20.3	16.7
14.7	11.7	7.9	14.5
8.6	8.9	12.5	15.3
11.5	6.1	7.2	10.4
	19.2		9.1

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.

(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					---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

treatment	I	II	III	IV
count, ni =	5	6	5	6
mean , x̅ i =	17.120	11.48	14.140	13.85
std. dev., si =	7.999	4.581	7.119	3.337
sample variances, si^2 =	63.992	20.986	50.683	11.135
total sum	85.6	68.9	70.7	83.1	308.3	(grand sum)
grand mean , x̅̅ =	Σni*x̅i/Σni =				14.01
square of deviation of sample mean from grand mean,( x̅ - x̅̅)²	9.649	6.402	0.016	0.027
					TOTAL
SS(between)= SSB = Σn( x̅ - x̅̅)² =	48.247	38.415	0.080	0.161	86.90258
SS(within ) = SSW = Σ(n-1)s² =	255.968	104.928	202.732	55.675	619.303

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 =    28.9675
  
mean square within groups , MSW = SSW/N-k =    34.4057
  
F-stat = MSB/MSW =    0.8419
----------------

b)

SS tot=   706.206  
SS bet=   86.903  
SSW=   619.303  
      
dfBET   =   3
dfW   =   18
MSBET   =   28.968
MSW   =   34.406
      
F=   0.842  

(numerator)=3
(denominator)=18

f)

anova table
	SS	df	MS	F	p-value
Between:	86.903	3	28.968	0.842	p-value > 0.100	Do not reject H0
Within:	619.303	18	34.406
Total:	706.206	21
					α =	0.05