Chapter 7.4, Problem 3E

Using Table G, find the values for χ² left and χ² right.

a. α = 0.05, n = 12

b. α = 0.10, n = 20

c. α = 0.05, n = 27

d. α = 0.01, n = 6

e. α = 0.10, n = 41

To determine

To find: The critical values ${\chi }_{\text{left}}^2$ and ${\chi }_{\text{right}}^2$ when $n=12$ and $\alpha =0.05$.

Answer to Problem 3E

The critical values ${\chi }_{\text{left}}^2$ and ${\chi }_{\text{right}}^2$ is 3.816 and 21.92, respectively.

Explanation of Solution

Given info:

$n=12$ and $\alpha =0.05$.

Calculation:

Degrees of freedom:

$\begin{array}{c}n-1=12-1\\ =11\end{array}$

Software Procedure:

Step-by-step procedure to obtain the critical value using the MINITAB software:

• Choose Graph > Probability Distribution Plot choose View Probability> OK.
• From Distribution, choose ‘chi square’ distribution.
• In Degrees of freedom, enter 11.
• Click the Shaded Area tab.
• Choose Probability value and Both Tail for the region of the curve to shade.
• Enter the Probability value as 0.05.
• Click OK.

Output using the MINITAB software is given below:

Thus, the critical values ${\chi }_{\text{left}}^2$ and ${\chi }_{\text{right}}^2$ is 3.816 and 21.92, respectively.

To determine

To find: The critical values ${\chi }_{\text{left}}^2$ and ${\chi }_{\text{right}}^2$ when $n=20$ and $\alpha =0.10$.

Answer to Problem 3E

The critical values ${\chi }_{\text{left}}^2$ and ${\chi }_{\text{right}}^2$ is 10.12 and 30.14, respectively.

Explanation of Solution

Given info:

$n=20$ and $\alpha =0.10$.

Calculation:

Degrees of freedom:

$\begin{array}{c}n-1=20-1\\ =19\end{array}$

Software Procedure:

Step-by-step procedure to obtain the critical value using the MINITAB software:

• Choose Graph > Probability Distribution Plot choose View Probability> OK.
• From Distribution, choose ‘chi square’ distribution.
• In Degrees of freedom, enter 19.
• Click the Shaded Area tab.
• Choose Probability value and Both Tail for the region of the curve to shade.
• Enter the Probability value as 0.10.
• Click OK.

Output using the MINITAB software is given below:

Thus, the critical values ${\chi }_{\text{left}}^2$ and ${\chi }_{\text{right}}^2$ is 10.12 and 30.14, respectively.

To determine

To find: The critical values ${\chi }_{\text{left}}^2$ and ${\chi }_{\text{right}}^2$ when $n=27$ and $\alpha =0.05$.

Answer to Problem 3E

The critical values ${\chi }_{\text{left}}^2$ and ${\chi }_{\text{right}}^2$ is 13.84 and 41.92, respectively.

Explanation of Solution

Given info:

$n=27$ and $\alpha =0.05$.

Calculation:

Degrees of freedom:

$\begin{array}{c}n-1=27-1\\ =26\end{array}$

Software Procedure:

Step-by-step procedure to obtain the critical value using the MINITAB software:

• Choose Graph > Probability Distribution Plot choose View Probability> OK.
• From Distribution, choose ‘chi square’ distribution.
• In Degrees of freedom, enter 26.
• Click the Shaded Area tab.
• Choose Probability value and Both Tail for the region of the curve to shade.
• Enter the Probability value as 0.05.
• Click OK.

Output using the MINITAB software is given below:

Thus, the critical values ${\chi }_{\text{left}}^2$ and ${\chi }_{\text{right}}^2$ is 13.84 and 41.92, respectively.

To determine

To find: The critical values ${\chi }_{\text{left}}^2$ and ${\chi }_{\text{right}}^2$ when $n=6$ and $\alpha =0.01$.

Answer to Problem 3E

The critical values ${\chi }_{\text{left}}^2$ and ${\chi }_{\text{right}}^2$ is 0.4117 and 16.75, respectively.

Explanation of Solution

Given info:

$n=6$ and $\alpha =0.01$.

Calculation:

Degrees of freedom:

$\begin{array}{c}n-1=6-1\\ =5\end{array}$

Software Procedure:

Step-by-step procedure to obtain the critical value using the MINITAB software:

• Choose Graph > Probability Distribution Plot choose View Probability> OK.
• From Distribution, choose ‘chi square’ distribution.
• In Degrees of freedom, enter 5.
• Click the Shaded Area tab.
• Choose Probability value and Both Tail for the region of the curve to shade.
• Enter the Probability value as 0.01.
• Click OK.

Output using the MINITAB software is given below:

Thus, the critical values ${\chi }_{\text{left}}^2$ and ${\chi }_{\text{right}}^2$ is 0.4117 and 16.75, respectively.

To determine

To find: The critical values ${\chi }_{\text{left}}^2$ and ${\chi }_{\text{right}}^2$ when $n=41$ and $\alpha =0.10$.

Answer to Problem 3E

The critical values ${\chi }_{\text{left}}^2$ and ${\chi }_{\text{right}}^2$ is 26.51 and 55.76, respectively.

Explanation of Solution

Given info:

$n=41$ and $\alpha =0.10$.

Calculation:

Degrees of freedom:

$\begin{array}{c}n-1=41-1\\ =40\end{array}$

Software Procedure:

Step-by-step procedure to obtain the critical value using the MINITAB software:

• Choose Graph > Probability Distribution Plot choose View Probability> OK.
• From Distribution, choose ‘chi square’ distribution.
• In Degrees of freedom, enter 40.
• Click the Shaded Area tab.
• Choose Probability value and Both Tail for the region of the curve to shade.
• Enter the Probability value as 0.10.
• Click OK.

Output using the MINITAB software is given below:

Thus, the critical values ${\chi }_{\text{left}}^2$ and ${\chi }_{\text{right}}^2$ is 26.51 and 55.76, respectively