Chapter 11, Problem 1PS

An experiment has a single factor with five groups and seven values in each group.

a. How many degrees of freedom are there is determining the among-group variation?

b. How many degrees of freedom are there in determining the within-group variation?

c. How many degrees of freedom are there in determining the total variation?

To determine

Determine the value of degrees of freedom for among group variation.

Answer to Problem 1PS

The required value of degrees of freedom is 4.

Explanation of Solution

In a study, among group variation is measured between groups of the data. The degree of freedom for among group variation can be calculated as:

$\text{Degrees of freedom}=c-1$

Here, $c$ is the number of groups present in the data set. Since 5 groups have been compared in the study, substitute $n=5$ in the above formula.

$\begin{array}{c}DF=c-1\\ =5-1\\ =4\end{array}$

Hence, the value of degrees of freedom for among group variation is $4$ .

To determine

Determine the value of degrees of freedom for within group variation.

Answer to Problem 1PS

The required value of degrees of freedom is 30.

Explanation of Solution

In a study, within group variation is measured between the data points of each group. The degree of freedom for within group variation can be calculated as:

$\text{Degrees of freedom}=n-c$

Here, $n$ is the total number of data values and $c$ is the number of groups.

Since, there are 5 groups with 7 values each, the total number of data values is:

$\left(5\times 7\right)=35$ .

Substitute $n=35$ in the above formula.

$\begin{array}{c}\text{DF}=n-c\\ =35-5\\ =30\end{array}$

Hence, the value of degrees of freedom for within group variation is $30$ .

To determine

Determine the value of number of degrees for the total variation.

Answer to Problem 1PS

The required value of degrees of freedom is 34.

Explanation of Solution

The degree of freedom for the total variation can be calculated as:

$\text{DF}=n-1$

Here, $n$ is the total number of data values. Since the total number of data values is 35, the value of degrees of freedom for the total variation is:

$\begin{array}{c}DF=n-1\\ =35-1\\ =34\end{array}$

Hence, the value of degrees of freedom is 34.