Question

In: Math

How productive are U.S. workers? One way to answer this question is to study annual profits...

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 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)=

(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

Solutions

Expert Solution

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

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