Chapter 13, Problem 39E

(a)

To determine

To find: The pooled estimate of the standard deviation with degree of freedom.

Answer to Problem 39E

Solution: The pooled estimate of the standard deviation is 38.14 with degree of freedom 105.

Explanation of Solution

Calculation: The pooled variance can be calculated as

$S_p=\sqrt{\frac{(n_1-1)s_1^2+(n_2-1)s_2^2+(n_3-1)s_3^2+(n_4-1)s_4^2}{n_1+n_2+n_3+n_4-4}}$

where $s_1^2,s_2^2,s_3^2,\text{ and }{s}_4^2$ are the sample variances and $n_1,n_2,n_3,\text{ and }{n}_4$ are the sample observations. Substitute the values in the formula above:

$\begin{array}{c}{S}_p=\sqrt{\frac{(32-1)\times {36.4}^2+(25-1)\times {31.2}^2+(25-1)\times {41.6}^2+(27-1)\times {42.4}^2}{32+25+25+27-4}}\\ =38.14\end{array}$

The degrees of freedom can be calculated as

$\begin{array}{c}df=n_1+n_2+n_3+n_4-4\\ =32+25+25+27-4\\ =105\end{array}$

(b)

To determine

To test: Whether it is appropriate to use a pooled standard deviation for the provided analysis.

Answer to Problem 39E

Solution: Yes, it is appropriate to use a pooled standard deviation for the provided analysis.

Explanation of Solution

Calculation: If the largest standard deviation of a group is less than the twice of the smallest standard deviation, then one can use the pooled standard deviation for the analysis. That is,

$\begin{array}{c}2s=2\times 31.2\\ =62.4\end{array}$.

Conclusion: The largest pooled standard deviation for the provided analysis is 42.4, which is less than twice the smallest pooled standard deviation of the provided analysis. Therefore, it is appropriate to use a pooled standard deviation for the provided analysis.

(c)

To determine

To find: The marginal means.

Answer to Problem 39E

Solution: The marginal means of sender individual is 70.9, for sender group is 48.85, for responder individual is 59.75, and for responder group is 60.

Explanation of Solution

Calculation: Marginal mean is calculated by the average of row or column of the provided table. The marginal means of sender individual can be calculated as

$\begin{array}{c}\text{Marginal mean}=\frac{\text{Individual}+\text{Group}}{2}\\ =\frac{65.5+76.3}{2}\\ =70.9\end{array}$

The marginal means of sender group can be calculated as

$\begin{array}{c}\text{Marginal mean}=\frac{\text{Individual}+\text{Group}}{2}\\ =\frac{54.0+43.7}{2}\\ =48.85\end{array}$

The marginal means of responder individual can be calculated as

$\begin{array}{c}\text{Marginal mean}=\frac{\text{Individual}+\text{Group}}{2}\\ =\frac{65.5+54.0}{2}\\ =59.75\end{array}$

The marginal means of responder group can be calculated as

$\begin{array}{c}\text{Marginal mean}=\frac{\text{Individual}+\text{Group}}{2}\\ =\frac{76.3+43.7}{2}\\ =60\end{array}$

Therefore, the marginal means of sender individual is 70.9, for sender group is 48.85, for responder individual is 59.75, and for responder group is 60.

(d)

To determine

To graph: The means.

Explanation of Solution

Calculation: To plot the means, use Minitab and follow the steps below:

Step 1: Open the Minitab worksheet.

Step 2: Go to ANOVA > Interaction plot.

Step 3: Select “Mean” in the column for “Responses” and select “Respondent and Sender” in the column for “Factors.”.

The plot for means is obtained as

Interpretation: The patterns are not parallel. It can be seen from the plot that there is an interaction among sender and responder. So, there is an interaction among group A and group B. This interaction is significant as two lines are not parallel to each other.

(e)

To determine

To find: P-values and its interpretation for the F-statistics.

Answer to Problem 39E

Solution: The sender effect is significant as P-value is less than 0.05 significance level. The responder effect is not significant as P- value is greater than 0.05. The interaction effect is not significant as P- value is greater than 0.05.

Explanation of Solution

Calculation: The degree of freedom for sender is calculated by

$\begin{array}{c}DF=\left(I-1\right)\\ =1\end{array}$

The degree of freedom for responder is calculated by

$\begin{array}{c}DF=\left(J-1\right)\\ =1\end{array}$

The degree of freedom for interaction is calculated by

$\begin{array}{c}DF=\left(J-1\right)\left(I-1\right)\\ =1\end{array}$

The degree of freedom for error term is calculated by

$\begin{array}{c}DF=N-IJ\\ =109-4\\ =105\end{array}$

The P-value for sender group can be calculated by using the formula $=FDIST\left(9.05,1,105\right)$. The screenshot of the used formula is shown below:

The P-value for responder group can be calculated by using the formula $\text{=FDIST(0}\text{.001,1,105)}$. The screenshot of the used formula is shown below:

The P-value for interaction can be calculated by using the formula $\text{=FDIST(2}\text{.08,1,105)}$. The screenshot of the used formula is shown below:

Interpretation: Therefore only the sender effect is significant as P-value is less than 0.05 significance level. The responder effect is not significant as P- value is greater than 0.05. The interaction effect is not significant as P- value is greater than 0.05.