Chapter 12, Problem 22P

To create the following data we started with the same sample means and variances that appeared in problem 21 but doubled the sample size to n = 10.

a. Predict how the increase in sample size should affect the F-ratio for these data compares to the values obtained in problem 21. Use an ANOVA with a – .05 to check your prediction. Note: Recause the sample are all the same size, MS_within is the average of the three sample variances.

b. Predict how the increase in sample size should affect the value of η’ for these data compared to the η’ in problem 21. Calculate η’ to check your prediction.

To determine

To predict: The effect on size of the F-ratio after increase in sample size.

To determine: There are any significant mean differences between treatments.

Answer to Problem 22P

On increasing the sample size, the size of F-ratio increases. There are significant mean differences between the treatments.

Explanation of Solution

Given info:

Treatment
I	II	III
$n=10$	$n=10$	$n=10$	$N=30$
$M=1$	$M=5$	$M=6$	$G=120$
$T=10$	$T=50$	$T=60$	$\sum X^2=890$
$s^2=9$	$s^2=10$	$s^2=11$
$SS=81$	$SS=90$	$SS=99$

Level of significance is $\alpha =0.05$.

Calculation:

On increasing the sample size, the sum of squares between treatments increases, hence numerator of the F-ratio increases. Therefore, size of F-ratio increases and chances to reject the null hypothesis increases.

Let k represents the numbers of treatments. Then,

$k=3$

Then, degrees of freedom corresponding to $S{S}_{between}$ are:

$\begin{array}{c}d{f}_{between}=k-1\\ =3-1\\ =2\end{array}$

Then, degrees of freedom corresponding to $S{S}_{within}$ are:

$\begin{array}{c}d{f}_{within}=N-k\\ =30-3\\ =27\end{array}$

$S{S}_{within}$ is calculated as:

$\begin{array}{c}S{S}_{within}=\sum SS\\ =81+90+99\\ =270\end{array}$

Sum of squares between treatments is given as:

$\begin{array}{c}S{S}_{between}=\sum \frac{T^2}{n}-\frac{G^2}{N}\\ =\frac{{\left(10\right)}^2}{10}+\frac{{\left(50\right)}^2}{10}+\frac{{\left(60\right)}^2}{10}-\frac{{\left(120\right)}^2}{30}\\ =10+250+360-480\\ =140\end{array}$

F-ratio is calculated as:

$\begin{array}{c}F=\frac{\frac{S{S}_{between}}{d{f}_{between}}}{\frac{S{S}_{within}}{d{f}_{within}}}\\ =\frac{\frac{140}{2}}{\frac{270}{27}}\\ =\frac{70}{10}\\ =7\end{array}$

From the table $B.4$ of appendix B in the textbook, critical value corresponding to $df=2,12$ is 3.88.

Since, F-ratio is greater than the critical value; therefore, the null hypothesis is rejected and conclude that there are significant mean differences between the treatments.

Conclusion:

On increasing the sample size, the size of F-ratio increases. There are significant mean differences between the treatments.

To determine

To predict: The effect of increase in sample size on the effect size ${\eta }^2$ for the study.

To determine: The effect size ${\eta }^2$ for the study.

Answer to Problem 22P

On increasing the sample size: the effect size ${\eta }^2$ for the study increases. The effect size ${\eta }^2$ for the study is 0.341.

Explanation of Solution

Calculation:

The formula for ${\eta }^2$ is

${\eta }^2=\frac{S{S}_{between}}{S{S}_{total}}$

On increasing the sample size, the sum of squares between treatments increases more rapidly than total sum of squares, so numerator becomes larger. Hence, the effect size ${\eta }^2$ for the study increases.

$S{S}_{within}$ is calculated as:

$\begin{array}{c}S{S}_{within}=\sum SS\\ =81+90+99\\ =270\end{array}$

Sum of squares between treatments is given as:

$\begin{array}{c}S{S}_{between}=\sum \frac{T^2}{n}-\frac{G^2}{N}\\ =\frac{{\left(10\right)}^2}{10}+\frac{{\left(50\right)}^2}{10}+\frac{{\left(60\right)}^2}{10}-\frac{{\left(120\right)}^2}{30}\\ =10+250+360-480\\ =140\end{array}$

Total sum of squares is calculated as:

$\begin{array}{c}S{S}_{total}=S{S}_{within}+S{S}_{between}\\ =270+140\\ =410\end{array}$

The effect size for the study is given as:

$\begin{array}{c}{\eta }^2=\frac{S{S}_{between}}{S{S}_{total}}\\ =\frac{140}{410}\\ =0.341\end{array}$

Conclusion:

On increasing the sample size: the effect size ${\eta }^2$ for the study increases. The effect size ${\eta }^2$ for the study is 0.341.