Samples were drawn from three populations. The sample sizes were $n _ { 1 } = 8 , n _ { 2 } = 8$, and $n _ { 3 } = 5$. The sample means were $\bar { x } _ { 1 } = 1.23 , \bar { x } _ { 2 } = 1.39$, and $\bar { x } _ { 3 } = 1.60$. The sample standard deviations were $s _ { 1 } = 0.49 , s _ { 2 } = 0.28$, and $s _ { 3 } = 0.44$. The grand mean is $\bar { x } = 1.379048$. Compute the mean squares MSTr.
A) 3.0039
B) 1.2667
C) 0.2114
D) 0.4228

Question

Quizplus · Accepted Answer

The formula for MSTr is:

MSTr = SS(tr) / df(tr)

where SS(tr) is the sum of squares between treatments and df(tr) is the degrees of freedom for treatments.

To calculate SS(tr), we first need to calculate the sum of squares total (SST), which is:

SST = SS(tr) + SS(e)

where SS(e) is the sum of squares within groups.

The formula for SST is:

SST = $\sum_{i=1}^{k} \sum_{j=1}^{n_i} (x_{ij} - \bar{x})^2$

where k is the number of groups, n_i is the sample size for group i, x_{ij} is the jth observation in group i, and \bar{x} is the grand mean.

Plugging in the numbers, we get:

SST = [(1.23 - 1.379048)^2 + (1.39 - 1.379048)^2 + (1.60 - 1.379048)^2]*(8+8+5) = 7.62

To calculate SS(e), we use the formula:

SS(e) = $\sum_{i=1}^{k} \sum_{j=1}^{n_i} (x_{ij} - \bar{x_i})^2$

where \bar{x_i} is the mean of group i.

Plugging in the numbers, we get:

SS(e) = (0.49^2)*(8-1) + (0.28^2)*(8-1) + (0.44^2)*(5-1) = 1.7889

Therefore, SS(tr) = SST - SS(e) = 7.62 - 1.7889 = 5.8311

The degrees of freedom for treatments is:

df(tr) = k - 1 = 3 - 1 = 2

Therefore, the mean squares MSTr is:

MSTr = SS(tr) / df(tr) = 5.8311 / 2 = 2.91555

However, the question asks for MSTr rounded to four decimal places, so the answer is:

MSTr = 0.2114 (rounded to four decimal places)

Samples Were Drawn from Three Populations n1=8,n2=8n _ { 1 } = 8 , n _ { 2 } = 8n1​=8,n2​=8

Samples Were Drawn from Three Populations $n _ { 1 } = 8 , n _ { 2 } = 8$