Chapter 14.1, Problem 3E

(a)

To determine

Compute the ANOVA table.

Explanation of Solution

Table -1 shows the statistics values as follows:

Table -1

Statistic	Treatment
	1	2	3	4
n	10	14	11	18
$\overline{\text{x}}$	30	35	33	40
${\text{s}}^2$	10	10	10	10

$\begin{array}{c}\overline{\text{x}}=\frac{10\left(30\right)+14\left(35\right)+11\left(33\right)+18\left(40\right)}{10+14+11+18}\\ =\frac{300+490+363+720}{53}\\ =\frac{1,873}{53}\\ =35.34\end{array}$

SST can be calculated as follows:

$\begin{array}{c}\text{SST}=\sum {\text{n}}_{\text{j}}\left(\overline{{\text{x}}_{\text{j}}}-\overline{\text{x}}\right)\\ =10{\left(30-35.34\right)}^2+14{\left(35-35.34\right)}^2+11\left(33-35.34\right)+18{\left(40-35.34\right)}^2\\ =737.9\end{array}$

SSE can be calculated as follows:

$\begin{array}{c}\text{SSE}=\sum \left({\text{n}}_{\text{j}}-1\right){\text{s}}_{\text{j}}^2\\ =\left(10-1\right)10+\left(14-1\right)10+\left(11-1\right)10+\left(18-1\right)10\\ =490\end{array}$

Thus, the value of SSE is 490

Table -2 shows the ANOVA Table as follows:

Table -2
Source	Degree of freedom	Sum of square	Mean square	F
Treatments	k-1 = 3	SST = 737.9	246	24.6
Error	n- k = 49	SSE = 490	10

(b)

To determine

Compute the ANOVA table if sample means is 130,135,133 and 140.

Explanation of Solution

$\begin{array}{c}\overline{\text{x}}=\frac{10\left(130\right)+14\left(135\right)+11\left(133\right)+18\left(140\right)}{10+14+11+18}\\ =\frac{1,300+1,890+1,463+2,520}{53}\\ =\frac{7,173}{53}\\ =135.34\end{array}$

SST can be calculated as follows:

$\begin{array}{c}\text{SST}=\sum {\text{n}}_{\text{j}}\left(\overline{{\text{x}}_{\text{j}}}-\overline{\text{x}}\right)\\ =10{\left(130-135.34\right)}^2+14{\left(135-135.34\right)}^2+11{\left(133-135.34\right)}^2+18{\left(140-135.34\right)}^2\\ =737.9\end{array}$

SSE can be calculated as follows:

$\begin{array}{c}\text{SSE}=\sum \left({\text{n}}_{\text{j}}-1\right){\text{s}}_{\text{j}}^2\\ =\left(10-1\right)10+\left(14-1\right)10+\left(11-1\right)10+\left(18-1\right)10\\ =610\end{array}$

Thus, the value of SSE is 610

Table -3 shows the ANOVA Table as follows:

Table -3
Source	Degree of freedom	Sum of square	Mean square	F
Treatments	k-1 = 3	SST = 737.9	246	24.60
Error	n- k = 49	SSE = 490	10

(c)

To determine

Explain the F- statistic

Explanation of Solution

From the above calculation, when the sample means changes, then there is no change in F- statistic.