Chapter 7.2, Problem 3E

Find the values for each.

a. t_α/2 and n = 18 for the 99% confidence interval for the mean

b. t_α/2 and n = 23 for the 95% confidence interval for the mean

c. t_α/2 and n = 15 for the 98% confidence interval for the mean

d. t_α/2 and n = 10 for the 90% confidence interval for the mean

e. t_α/2 and n = 20 for the 95% confidence interval for the mean

To determine

To find: The critical value $t_{\frac{\alpha }{2}}$.

Answer to Problem 3E

The critical value $t_{\frac{\alpha }{2}}$ is2.898.

Explanation of Solution

Given info:

$n=18$ for the 99% confidence interval for the mean.

Calculation:

Degrees of freedom:

$\begin{array}{c}df=n-1\\ =18-1\\ =17\end{array}$

Critical value:

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 ‘t’ distribution.
• In Degrees of freedom, enter 17.
• 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 critical value $\left(t_{\frac{\alpha }{2}}\right)$ is 2.898.

To determine

To find: The critical value $t_{\frac{\alpha }{2}}$.

Answer to Problem 3E

The critical value $t_{\frac{\alpha }{2}}$ is 2.074.

Explanation of Solution

Given info:

$n=23$ for the 95% confidence interval for the mean.

Calculation:

Degrees of freedom:

$\begin{array}{c}df=n-1\\ =23-1\\ =22\end{array}$

Critical value:

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 ‘t’distribution.
• In Degrees of freedom, enter 22.
• 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 critical value $\left(t_{\frac{\alpha }{2}}\right)$ is 2.074.

To determine

To find: The critical value $t_{\frac{\alpha }{2}}$.

Answer to Problem 3E

The critical value $t_{\frac{\alpha }{2}}$ is 2.624.

Explanation of Solution

Given info:

$n=15$ for the 98% confidence interval for the mean.

Calculation:

Degrees of freedom:

$\begin{array}{c}df=n-1\\ =15-1\\ =14\end{array}$

Critical value:

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 ‘t’distribution.
• In Degrees of freedom, enter 14.
• 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 critical value $\left(t_{\frac{\alpha }{2}}\right)$ is 2.624.

To determine

To find: The critical value $t_{\frac{\alpha }{2}}$.

Answer to Problem 3E

The critical value $t_{\frac{\alpha }{2}}$ is 1.833.

Explanation of Solution

Given info:

$n=10$ for the 90% confidence interval for the mean.

Calculation:

Degrees of freedom:

$\begin{array}{c}df=n-1\\ =10-1\\ =9\end{array}$

Critical value:

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 ‘t’ distribution.
• In Degrees of freedom, enter 9.
• 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 critical value $\left(t_{\frac{\alpha }{2}}\right)$ is 1.833.

To determine

To find: The critical value $t_{\frac{\alpha }{2}}$.

Answer to Problem 3E

The critical value $t_{\frac{\alpha }{2}}$ is 2.093.

Explanation of Solution

Given info:

$n=20$ for the 95% confidence interval for the mean.

Calculation:

Degrees of freedom:

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

Critical value:

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 ‘t’ 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 0.05.
• Click OK.

Output using the MINITAB software is given below:

From the output, the critical value $\left(t_{\frac{\alpha }{2}}\right)$ is 2.093.