Samples were drawn from three populations. The sample sizes were $n _ { 1 } = 5 , n _ { 2 } = 5$
$n _ { 3 } = 8$ The sample means were $\bar { x } _ { 1 } = 1.65 , \bar { x } _ { 2 } = 1.70 , \text { and } \bar { x } _ { 3 } = 1.41$ the sample standard deviations were $s _ { 1 } = 0.44 , s _ { 2 } = 0.25$ and $s _ { 3 } = 0.26$ The grand mean is $\overline { \bar { x } } = 1.557222$

Compute the sum of squares SSTr.
A) 1.4976
B) 0.3184
C) 1.5944
D) 0.1592

Question

Samples were drawn from three populations. The sample sizes were $n _ { 1 } = 5 , n _ { 2 } = 5$
$n _ { 3 } = 8$ The sample means were $\bar { x } _ { 1 } = 1.65 , \bar { x } _ { 2 } = 1.70 , \text { and } \bar { x } _ { 3 } = 1.41$  the sample standard deviations were $s _ { 1 } = 0.44 , s _ { 2 } = 0.25$ and $s _ { 3 } = 0.26$ The grand mean is $\overline { \bar { x } }  = 1.557222$

Compute the sum of squares SSTr. 
A) 1.4976 
B)  0.3184
C) 1.5944 
D) 0.1592

Quizplus · Accepted Answer

The sum of squares between treatments (SSTr) is calculated using the formula: $$ SSTr = \sum_{i=1}^{k} n_i (\bar{x}_i - \overline{\bar{x}})^2 $$ where $n_i$ is the sample size for group $i$, $\bar{x}_i$ is the sample mean for group $i$, and $\overline{\bar{x}}$ is the grand mean. Plugging in the given values: $$ SSTr = 5(1.65 - 1.557222)^2 + 5(1.70 - 1.557222)^2 + 8(1.41 - 1.557222)^2 $$ $$ SSTr = 5(0.092778)^2 + 5(0.142778)^2 + 8(-0.147222)^2 $$ $$ SSTr = 5(0.0086) + 5(0.0204) + 8(0.0217) $$ $$ SSTr = 0.043 + 0.102 + 0.1736 = 0.3186 $$ Rounding to four decimal places gives 0.3184.

Samples Were Drawn from Three Populations n1=5,n2=5n _ { 1 } = 5 , n _ { 2 } = 5n1​=5,n2​=5

Samples Were Drawn from Three Populations $n _ { 1 } = 5 , n _ { 2 } = 5$