Chapter 12, Problem 42E

(a)

To determine

To find: The $F$ -statistic value for a pre specified p-value.

Answer to Problem 42E

Solution: The required $F$ -statistic value is 4.82.

Explanation of Solution

Calculation: Follow the below described steps to compute the F-statistic value using Applet,

Step 1: To compute the F-statistic value select the option ‘Statistical Applets: One-Way Anova’ as shown in the screenshot below,

Step 2: The ANOVA window is displayed in the screenshot below,

Step 3: In the ANOVA window that appears drag the black dots until the p-value at the lower side of the window is set as 0.01. On moving the black dots, the corresponding means, standard deviations, $F$ -statistic and the p-value changes.

On a few configurations, the p-value is set at $0.01$ shown below.

As is evident from the applet, the $F$ -statistic value is $4.82$.

(b)

To determine

To find: The changes in $F$ -statistic and p-value on increasing the variations within the groups.

Answer to Problem 42E

Solution: The $F$ -statistic value decreases while the p-value keeps on increasing.

Explanation of Solution

Calculation: Repeat Step 1 of part (a) and on sliding the standard deviations bar to the right, it is observed that there is a gradual drop in the $F$ -statistic value and hike in the p-value.

The corresponding $F$ -statistic value is $12.12$ and the p-value is $<0.0001$.

On sliding the standard deviation bar further to the right side, the following changes can be observed

The $F$ - statistic value drops down from $12.12$ to $6.00$. The p-value escalates to $0.0034$.

On further adjustments,

Interpretation: The $F$ - statistic value further drops down to from $6.00$ to $2.72$. There is a considerable hike in the p-value from $0.0034$ to $0.0706$.

(c)

To determine

To explain: The reason for a change in $F$ - statistic value and p-value based on the concepts of variations with the groups and between the groups.

Answer to Problem 42E

Solution: The $F$ -statistic value is large for higher variations between the groups. It is small in case of higher variability within the groups. The p-value is large in case of between variability and is not quite helpful in case of large within variability.

Explanation of Solution

The ANOVA technique is the analysis of variations within the groups and between the groups. The extent of variation is measured by F-statistic. In simple words, F statistic is the ratio of variations between the group and variations within groups.

An increase in within the group variations leads to a drop in the $F$ - statistic value.

Considering the p-value, in case of variations between the groups, the p-value tends to be high accounting for a significant difference between the groups. In case of within variability if the variability is quite large, then p-value does not yield any relevant inference.