Chapter 13, Problem 31E

(a)

To determine

To graph: The plot for means.

Explanation of Solution

Graph: To obtain the interaction plot, use Minitab. Follow the steps below:

Step 1: Insert the data into Minitab worksheet.

Step 2: Go to Stat > ANOVA > Interaction plot.

Step 3: Select “Means” in the column for Responses and select “Location and Car” in the column for Factors.

Step 4: Click OK.

The obtained interaction plot is obtained as shown below:

Interpretation: The interaction plot represents that there is a little interaction between two Chromium means. There may be an interaction for the larger values.

(b)

To determine

To find: The pooled standard deviation.

Answer to Problem 31E

Solution: The required pooled standard deviation is 0.12777.

Explanation of Solution

Calculation: The pooled standard deviation is calculated by using the formula as shown below:

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

The pooled standard deviation is obtained as follows:

$\begin{array}{c}{s}_p^{}=\sqrt{\frac{\left(n-1\right)\left(s_1^2+s_2^2+s_3^2+s_4^2\right)}{4\left(n-1\right)}}\\ =\sqrt{\frac{\left(s_1^2+s_2^2+s_3^2+s_4^2\right)}{4}}\end{array}$

$\begin{array}{c}=\sqrt{\frac{{0.12}^2+{0.12}^2+{0.14}^2+{0.13}^2}{4}}\\ =0.12777\end{array}$

Interpretation: The pooling of standard deviation can be applied here because the data values satisfies the condition that the value of maximum standard deviation is less than the twice of the minimum standard deviation value, i.e., ${\sigma }_{\max }<2\times {\sigma }_{\min }$.

(c)

Section 1:

To determine

The contrast to compare the mean for new with old car.

Answer to Problem 31E

Solution: The contrast of compared groups is as follows:

$c_1=1/2\times \left({\mu }_{New-city}+{\mu }_{New-highway}\right)-1/2\times \left({\mu }_{Old-city}+{\mu }_{Old-highway}\right)$

Explanation of Solution

A contrast (c) is defined as a combination of population means, $c={\displaystyle \sum _{i=1}^n{a}_i\times {\mu }_i}$ where $a_i$ is chosen as ${\displaystyle \sum _{i=1}^n{a}_i}=0$ and ${\mu }_i$ is the sample means. The contrast to compare the groups of interest can be obtained as:

$c_1=1/2\times \left({\mu }_{New-city}+{\mu }_{New-highway}\right)-1/2\times \left({\mu }_{Old-city}+{\mu }_{Old-highway}\right)$

Section 2:

To determine

The contrast to compare the means for new car while driving in city.

Answer to Problem 31E

Solution: The contrast of compared groups is as follows:

$c_2=1\times \left({\mu }_{New-city}\right)-1\times \left({\mu }_{New-Highway}\right)$

Explanation of Solution

A contrast (c) is defined as a combination of population means, $c={\displaystyle \sum _{i=1}^n{a}_i\times {\mu }_i}$ where $a_i$ is chosen as ${\displaystyle \sum _{i=1}^n{a}_i}=0$ and ${\mu }_i$ is the sample means. The contrast to compare the groups of interest can be obtained as:

$c_2=1\times \left({\mu }_{New-city}\right)-1\times \left({\mu }_{New-Highway}\right)$

Section 3:

To determine

The contrast to compare the mean for highway with city while driving an old car.

Answer to Problem 31E

Solution: The contrast of compared groups is as follows:

$c_3=1\times \left({\mu }_{Old-highway}\right)-1\times \left({\mu }_{Old-city}\right)$

Explanation of Solution

A contrast (c) is defined as a combination of population means, $c={\displaystyle \sum _{i=1}^n{a}_i\times {\mu }_i}$ where $a_i$ is chosen as ${\displaystyle \sum _{i=1}^n{a}_i}=0$ and ${\mu }_i$ is the sample means. The contrast to compare the groups of interest can be obtained as:

$c_3=1\times \left({\mu }_{Old-highway}\right)-1\times \left({\mu }_{Old-city}\right)$

(d)

To determine

The benefit of repeated measurements.

Answer to Problem 31E

Solution: The repeated measure of study reduces the within group variation.

Explanation of Solution

The benefit of observing the group of participants in repeated manner rather than observing the random participants in each treatment reduces the within-group variation or within group errors. It is observed that by subjecting the same individuals to four treatments, rather than four individuals to one treatment each, the group within group’s variability is reduced. The repeated measure of study reduces the within-group variability which improves the F-statistic to define the significance of test.