Chapter 8.1, Problem 11E

a.

To determine

To find: The critical value $\left(z_{\frac{\alpha }{2}}\right)$ for the level of significance $\alpha =0.05$.

Answer to Problem 11E

The critical value $\left(z_{\frac{\alpha }{2}}\right)$ for the level of significance $\alpha =0.05$ is $\underset{\_}{\pm 1.96}$

Explanation of Solution

Given info:

$\alpha =0.05$ and two tailed test.

Calculation:

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 Normal.
• 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 value is $\underset{\_}{\pm 1.96}$

Thus, the critical value $\left(z_{\frac{\alpha }{2}}\right)$ for the level of significance $\alpha =0.05$ is $\underset{\_}{\pm 1.96}$

b.

To determine

To find: The left tailed critical value of z for the level of significance $\alpha =0.01$.

Answer to Problem 11E

The left tailed critical value of z for the level of significance $\alpha =0.01$ is $\left(z_{\alpha }\right)=-2.33$

Explanation of Solution

Given info:

$\alpha =0.01$ and left tailed test.

Calculation:

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 Normal.
• Under shaded area, select Define shaded area by Probability-value.
• Enter the Probability-value as 0.01.
• 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 -2.33.

Thus, the left tailed critical value of z for the level of significance $\alpha =0.01$ is $\left(-z_{\alpha }\right)=-2.33$.

c.

To determine

To find: The right tailed critical value of z for the level of significance $\alpha =0.005$.

Answer to Problem 11E

The right tailed critical value of z for the level of significance $\alpha =0.005$ is $\left(z_{\alpha }\right)=2.58$

Explanation of Solution

Given info:

$\alpha =0.005$ and right tailed test.

Calculation:

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 Normal.
• Under shaded area, select Define shaded area by Probability-value.
• Enter the Probability-value as 0.005.
• 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 2.58.

Thus, the right tailed critical value of z for the level of significance $\alpha =0.005$ is $\left(z_{\alpha }\right)=2.58$.

d.

To determine

To find: The right tailed critical value of z for the level of significance $\alpha =0.01$.

Answer to Problem 11E

The right tailed critical value of z for the level of significance $\alpha =0.01$ is $\left(z_{\alpha }\right)=2.33$

Explanation of Solution

Given info:

$\alpha =0.01$ and right tailed test.

Calculation:

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 Normal.
• Under shaded area, select Define shaded area by Probability-value.
• Enter the Probability-value as 0.01.
• 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 2.33.

Thus, the right tailed critical value of z for the level of significance $\alpha =0.01$ is $\left(z_{\alpha }\right)=2.33$.

e.

To determine

To find: The left tailed critical value of z for the level of significance $\alpha =0.05$.

Answer to Problem 11E

The left tailed critical value of z for the level of significance $\alpha =0.05$ is $\left(z_{\alpha }\right)=-1.645$

Explanation of Solution

Given info:

$\alpha =0.05$ and left tailed test.

Calculation:

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 Normal.
• Under shaded area, select Define shaded area by Probability-value.
• Enter the Probability-value as 0.05.
• 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 -1.645.

Thus, the left tailed critical value of z for the level of significance $\alpha =0.05$ is $\left(-z_{\alpha }\right)=-1.645$.