Chapter 7.1, Problem 7E

Find each.

a. z_α/2 for the 99% confidence interval

b. z_α/2 for the 98% confidence interval

c. z_α/2 for the 95% confidence interval

d. z_α/2 for the 90% confidence interval

e. z_α/2 for the 94% confidence interval

To determine

To find: The value of $z_{\frac{\alpha }{2}}$ for the 99% confidence interval.

Answer to Problem 7E

The value of $z_{\frac{\alpha }{2}}$ for the 99% confidence interval is 2.576.

Explanation of Solution

Given info:

The confidence level is 99%.

Calculation:

Since the 99% confidence interval is to be used.

$\begin{array}{c}\alpha =1-0.99\\ =0.01\\ \frac{\alpha }{2}=\frac{0.01}{2}\\ =0.005\end{array}$

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 ‘Normal’ distribution.
• 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:

From the output, the value of $z_{\frac{\alpha }{2}}$ for the 99% confidence interval is 2.576.

To determine

To find: The value of $z_{\frac{\alpha }{2}}$ for the 98% confidence interval.

Answer to Problem 7E

The value of $z_{\frac{\alpha }{2}}$ for the 98% confidence interval is 2.326.

Explanation of Solution

Given info:

The confidence level is 98%.

Calculation:

Critical value:

Since the 98% confidence interval is to be used.

$\begin{array}{c}\alpha =1-0.98\\ =0.02\\ \frac{\alpha }{2}=\frac{0.02}{2}\\ =0.01\end{array}$

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 ‘Normal’ distribution.
• Click the Shaded Area tab.
• Choose Probability Value and Both Tail for the region of the curve to shade.
• Enter the Probability value as 0.02.
• Click OK.

Output using the MINITAB software is given below:

From the output, the value of $z_{\frac{\alpha }{2}}$ for the 99% confidence interval is 2.326.

To determine

To find: The value of $z_{\frac{\alpha }{2}}$ for the 95% confidence interval.

Answer to Problem 7E

The value of $z_{\frac{\alpha }{2}}$ for the 95% confidence interval is 1.96.

Explanation of Solution

Given info:

The confidence level is 95%.

Calculation:

Critical value:

Since the 95% confidence interval is to be used.

$\begin{array}{c}\alpha =1-0.95\\ =0.05\\ \frac{\alpha }{2}=\frac{0.05}{2}\\ =0.025\end{array}$

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 ‘Normal’ distribution.
• 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:

From the output, the value of $z_{\frac{\alpha }{2}}$ for the 95% confidence interval is 1.96.

To determine

To find: The value of $z_{\frac{\alpha }{2}}$ for the 90% confidence interval.

Answer to Problem 7E

The value of $z_{\frac{\alpha }{2}}$ for the 90% confidence interval is 1.645.

Explanation of Solution

Given info:

The confidence level is 90%.

Calculation:

Critical value:

Since the 90% confidence interval is to be used.

$\begin{array}{c}\alpha =1-0.90\\ =0.10\\ \frac{\alpha }{2}=\frac{0.10}{2}\\ =0.05\end{array}$

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 ‘Normal’ distribution.
• 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:

From the output, the value of $z_{\frac{\alpha }{2}}$ for the 90% confidence interval is 1.645.

To determine

To find: The value of $z_{\frac{\alpha }{2}}$ for the 94% confidence interval.

Answer to Problem 7E

The value of $z_{\frac{\alpha }{2}}$ for the 94% confidence interval is 1.881.

Explanation of Solution

Given info:

The confidence level is 94%.

Calculation:

Critical value:

Since the 94% confidence interval is to be used.

$\begin{array}{c}\alpha =1-0.94\\ =0.06\\ \frac{\alpha }{2}=\frac{0.06}{2}\\ =0.03\end{array}$

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 ‘Normal’ distribution.
• Click the Shaded Area tab.
• Choose Probability Value and Both Tail for the region of the curve to shade.
• Enter the Probability value as 0.06.
• Click OK.

Output using the MINITAB software is given below:

From the output, the value of $z_{\frac{\alpha }{2}}$ for the 94% confidence interval is 1.1881.