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1. One-way ANOVA method example ( Enter your problem )
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1. Example-1

1. Solve using One-way ANOVA method
 Observation A B C D E 1 8 10 12 8 7 2 12 11 9 14 4 3 18 12 16 6 8 4 13 9 12 16 15

Solution:
 A B C D 8 12 18 13 10 11 12 9 12 9 16 12 8 14 6 16 7 4 8 15 sum A=45 sum B=50 sum C=60 sum D=65

 A^2 B^2 C^2 D^2 64 144 324 169 100 121 144 81 144 81 256 144 64 196 36 256 49 16 64 225 sum A^2=421 sum B^2=558 sum C^2=824 sum D^2=875

Data table
 Group A B C D Total N n_1=5 n_2=5 n_3=5 n_4=5 n=20 sum x_i T_1=sum x_1=45 T_2=sum x_2=50 T_3=sum x_3=60 T_4=sum x_4=65 sum x=220 sum x_(i)^2 sum x_1^2=421 sum x_2^2=558 sum x_3^2=824 sum x_4^2=875 sum x^2=2678 Mean bar x_i bar x_1=9 bar x_2=10 bar x_3=12 bar x_4=13 Overall bar x=11 Std Dev S_i S_1=2 S_2=3.8079 S_3=5.099 S_4=2.7386

Let k = the number of different samples = 4
n=n_1+n_2+n_3+n_4=5+5+5+5=20

Overall bar x=220/20=11

sum x=T_1+T_2+T_3+T_4=45+50+60+65=220 ->(1)

(sum x)^2/n=220^2/20=2420 ->(2)

sum T_i^2/n_i=(45^2/5+50^2/5+60^2/5+65^2/5)=2470 ->(3)

sum x^2=sum x_(1)^2+sum x_(2)^2+sum x_(3)^2+sum x_(4)^2=421+558+824+875=2678 ->(4)

ANOVA:
Step-1 : sum of squares between samples
"SSB"= (sum T_i^2/n_i) - (sum x)^2/n = (3)-(2)

=2470-2420

=50

Or
"SSB"=sum n_j * (bar x_j - bar x)^2

=5xx(9-11)^2+5xx(10-11)^2+5xx(12-11)^2+5xx(13-11)^2

=50

Step-2 : sum of squares within samples
"SSW"= sum x^2 - (sum T_i^2/n_i) = (4)-(3)

=2678-2470

=208

Step-3 : Total sum of squares
"SST"="SSB"+"SSW"

=50+208

=258

Step-4 : variance between samples
"MSB"=("SSB")/(k-1)

=50/(3)

=16.6667

Step-5 : variance within samples
"MSW"=("SSW")/(n-k)

=208/(20-4)

=208/(16)

=13

Step-6 : test statistic F for one way ANOVA test
F=("MSB")/("MSW")

=16.6667/(13)

=1.2821

the degree of freedom between samples
k-1=3

Now, degree of freedom within samples
n-k=20-4=16

ANOVA table
 Source of Variation Sums of SquaresSS Degrees of freedomDF Mean SquaresMS F Between samples SSB = 50 k-1 = 3 MSB = 16.6667 1.2821 Within samples SSW = 208 n-k = 16 MSW = 13 Total SST = 258 n-1 = 19

This material is intended as a summary. Use your textbook for detail explanation.
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