Chapter 8.5, Problem 1E

a.

To determine

To find: The critical value of ${\chi }^2$ for the right tailed test.

To state: The null and alternative hypothesis for the given situation.

Answer to Problem 1E

The critical value of ${\chi }^2$ for the right tailed test is ${\chi }_{\alpha }{}^2=27.59$

Explanation of Solution

Given info:

$\alpha =0.05$ $n=18$ and ${\sigma }^2=225$.

Calculation:

The test hypotheses:

Null hypothesis $H_0$:

$H_0:{\sigma }^2=225$

Alternative hypothesis $H_1$:

$H_1:{\sigma }^2>225$

Critical value:

The critical value can be obtained using the MINITAB software.

Statistical procedure:

The step by step procedure to find the critical-value using the MINITAB software is given below:

• Select Graphs, then Click on View probability>click ok.
• Under Distribution, choose Chi-square.
• Enter degrees of freedom as 17.
• Under shaded area, select Define shaded area by Probability-value.
• Enter the Probability-value as 0.05.
• Select right tail and click OK.

The output obtained by MINITAB is given below:

From the MINITAB output, it can be seen that the critical value is 27.59

Thus, the critical value of ${\chi }^2$ for the right tailed test is ${\chi }_{\alpha }{}^2=27.59$

b.

To determine

To find: The critical value of ${\chi }^2$ for the left tailed test.

To state: The null and alternative hypothesis for the given situation.

Answer to Problem 1E

The critical value of ${\chi }^2$ for the left tailed test is ${\chi }_{\left(1-\alpha \right)}{}^2=14.04$

Explanation of Solution

Given info:

$\alpha =0.1$ $n=23$ and ${\sigma }^2=225$.

Calculation:

The test hypotheses:

Null hypothesis $H_0$:

$H_0:{\sigma }^2=225$

Alternative hypothesis $H_1$:

$H_1:{\sigma }^2<225$

Critical value:

The critical value can be obtained using the MINITAB software.

Statistical procedure:

The step by step procedure to find the critical-value using the MINITAB software is given below:

• Select Graphs, then Click on View probability>click ok.
• Under Distribution, choose Chi-square.
• Enter degrees of freedom as 22.
• Under shaded area, select Define shaded area by Probability-value.
• Enter the Probability-value as 0.10.
• Select left tail and click OK.

The output obtained by MINITAB is given below:

From the MINITAB output, it can be seen that the critical value is 14.04

Thus, the critical value of ${\chi }^2$ for the left tailed test is ${\chi }_{\left(1-\alpha \right)}{}^2=14.04$

c.

To determine

To find: The critical value of ${\chi }^2$ for the two tailed test.

To state: The null and alternative hypothesis for the given situation.

Answer to Problem 1E

The critical values of ${\chi }^2$ for the two tailed test are ${\chi }_{\left(\frac{\alpha }{2}\right)}{}^2=26.12$ and ${\chi }_{\left(1-\frac{\alpha }{2}\right)}{}^2=5.629$

Explanation of Solution

Given info:

$\alpha =0.05$ $n=15$ and ${\sigma }^2=225$.

Calculation:

The test hypotheses:

Null hypothesis $H_0$:

$H_0:{\sigma }^2=225$

Alternative hypothesis $H_1$:

$H_1:{\sigma }^2\ne 225$

Critical value:

The critical value can be obtained using the MINITAB software.

Statistical procedure:

The step by step procedure to find the critical-value using the MINITAB software is given below:

• Select Graphs, then Click on View probability>click ok.
• Under Distribution, choose Chi-square.
• Enter degrees of freedom as 14.
• Under shaded area, select Define shaded area by Probability-value.
• Enter the Probability-value as 0.05.
• Select two tail and click OK.

The output obtained by MINITAB is given below:

From the MINITAB output, it can be seen that the critical values are 5.629 and 26.12.

Thus, the critical values of ${\chi }^2$ for the two tailed test are ${\chi }_{\left(\frac{\alpha }{2}\right)}{}^2=26.12$ and ${\chi }_{\left(1-\frac{\alpha }{2}\right)}{}^2=5.629$

d.

To determine

To find: The critical value of ${\chi }^2$ for the two tailed test.

To state: The null and alternative hypothesis for the given situation.

Answer to Problem 1E

The critical values of ${\chi }^2$ for the two tailed test are ${\chi }_{\left(\frac{\alpha }{2}\right)}{}^2=14.07$ and ${\chi }_{\left(1-\frac{\alpha }{2}\right)}{}^2=2.167$

Explanation of Solution

Given info:

$\alpha =0.10$ $n=8$ and ${\sigma }^2=225$.

Calculation:

The test hypotheses:

Null hypothesis $H_0$:

$H_0:{\sigma }^2=225$

Alternative hypothesis $H_1$:

$H_1:{\sigma }^2\ne 225$

Critical value:

The critical value can be obtained using the MINITAB software.

Statistical procedure:

The step by step procedure to find the critical-value using the MINITAB software is given below:

• Select Graphs, then Click on View probability>click ok.
• Under Distribution, choose Chi-square.
• Enter degrees of freedom as 7.
• Under shaded area, select Define shaded area by Probability-value.
• Enter the Probability-value as 0.10.
• Select two tail and click OK.

The output obtained by MINITAB is given below:

From the MINITAB output, it can be seen that the critical values are 2.167 and 14.07

Thus, the critical values of ${\chi }^2$ for the two tailed test are ${\chi }_{\left(\frac{\alpha }{2}\right)}{}^2=14.07$ and

${\chi }_{\left(1-\frac{\alpha }{2}\right)}{}^2=2.167$