Question

1. calculate ( ss_{\text{total}}, ss_{\text{between}} ), and ( ss_{\text{within}} ) for the following set of data:

treatment 1

treatment 2

treatment 3

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( n = 12 )

( n = 12 )

( n = 12 )

( n = 36 )

( t = 60 )

( t = 72 )

( t = 24 )

( g = 156 )

( ss = 30 )

( ss = 46 )

( ss = 40 )

( sum x^{2}=896 )

Explanation:

Step1: Calculate \(SS_{total}\)

Use the formula \(SS_{total}=\sum X^{2}-\frac{G^{2}}{N}\).
Substitute \(\sum X^{2} = 896\), \(G = 156\), and \(N = 36\) into the formula:
\(SS_{total}=896-\frac{156^{2}}{36}\)
First calculate \(\frac{156^{2}}{36}=\frac{24336}{36}=676\)
Then \(SS_{total}=896 - 676=220\)

Step2: Calculate \(SS_{between}\)

Use the formula \(SS_{between}=\sum\frac{T^{2}}{n}-\frac{G^{2}}{N}\)
For Treatment 1: \(\frac{T_{1}^{2}}{n}=\frac{60^{2}}{12}=\frac{3600}{12} = 300\)
For Treatment 2: \(\frac{T_{2}^{2}}{n}=\frac{72^{2}}{12}=\frac{5184}{12}=432\)
For Treatment 3: \(\frac{T_{3}^{2}}{n}=\frac{24^{2}}{12}=\frac{576}{12} = 48\)
\(\sum\frac{T^{2}}{n}=300 + 432+48=780\)
Then \(SS_{between}=780-\frac{156^{2}}{36}\)
Since \(\frac{156^{2}}{36}=676\), \(SS_{between}=780 - 676=104\)

Step3: Calculate \(SS_{within}\)

Use the formula \(SS_{within}=\sum SS_{each\ group}\)
Given \(SS_{1}=30\), \(SS_{2}=46\), \(SS_{3}=40\)
\(SS_{within}=30 + 46+40=116\)

Answer:

\(SS_{total} = 220\), \(SS_{between}=104\), \(SS_{within}=116\)