Description
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Explanation & Answer
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Solutions
Question 2
A
B
C
AB
AC
BC
ABC
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Total sum of replication
(1) = 78
a = 104
b = 119
ab =148
c = 127
ac =113
bc =164
abc =127
Average effect of: factor A
𝐴 = 𝑦̅𝐴+ − 𝑦̅𝐴− =
(𝑎+𝑎𝑏+𝑎𝑐+𝑎𝑏𝑐)
4𝑛
-
(1)+𝑏+𝑐+𝑏𝑐
4𝑛
= 1/4n (a + ab + ac + abc – (1) – b – c – bc)
=
1
4×3
(104 + 148 + 113 + 127 – 78 – 119 -127 – 164)
=1/12 (4) = 0.3333
Average effect of: factor B
𝐵 = 𝑦̅𝐵+ − 𝑦̅𝐵− =
=
1
(𝑏+𝑎𝑏+𝑏𝑐+𝑎𝑏𝑐)
4𝑛
-
(1)+𝑎+ 𝑐+𝑎𝑐
4𝑛
((119 +148 + 164 + 127) – ( 78 +104 + 127 + 113))
4×3
= 1/12 (136) = 11.3333
Average effect of: factor C.
𝐶 = 𝑦̅𝐶 + − 𝑦̅𝐶 − =
(𝑐+𝑎𝑐+𝑏𝑐+𝑎𝑏𝑐)
4𝑛
-
(1)+𝑎+ 𝑏+𝑎𝑏
4𝑛
1
= 4×3 (( 127 + 113 + 164 + 127 ) – (78 + 104 + 119 + 148 ))
= 1/12 (82) = 6.8333
Two- factor interaction effect between:
Factor A and B
AB =
=
(𝑎𝑏+𝑎𝑏𝑐−𝑎−𝑎𝑐)
4𝑛
(𝑏+𝑏𝑐−(1)−𝑐)
-
(148+127−104−113)
4𝑛
(119+164−78−127)
-
4∗3
4∗3
= 1/12 ((148+127+78+127) – (104 + 113 +119+164))
= 1/12 (-20)
= -1.6667
Factor A and C
AC =
=
(𝑎𝑐+𝑎𝑏𝑐−𝑎−𝑎𝑏)
4𝑛
(𝑐+𝑏𝑐−(1)−𝑏)
-
(113+127−104−148)
4𝑛
(127+164−78−119)
-
4∗3
4∗3
= 1/12 ((113+127+78+119) – (104+ 148 +127+164))
= 1/12 (-106)
= -8.3333
Factor B and C
BC =
=
(𝑏𝑐+𝑎𝑏𝑐−𝑏−𝑎𝑏)
4𝑛
(𝑐+𝑎𝑐−(1)−𝑎)
-
(164+127−119−148)
4∗3
4𝑛
(127+113−78−104)
-
4∗3
=1/12 ((164+127+78+104) – (119+148+127+113)
=1/12 (-34)
= -2.8333
Three factor interaction between:
Factors A, B and C
BC =
=
(𝑎𝑏𝑐−𝑏𝑐)−(𝑎𝑐−𝑐)
4𝑛
-
(127−164)−(113−127)
4∗3
(𝑎𝑏−𝑏)−(𝑎−(1)
-
4𝑛
(148−119)−(104−78)
4∗3
=1/12 ((127+104+119+127) – (148+113+164+78))
= 1/12 (-26)
= -2.1667
Let Г be the sum of the numerator for every factor.
Then,
The sum of squares associated with every factor is given by;
Г2
𝑆𝑆𝑖 = 8𝑛𝑖
Г2
𝐴
𝑆𝑆𝐴 = 8∗3
42
=
8∗3
= 0.6667
Г2
𝐵
𝑆𝑆𝐵 = 8∗3
=...
