Chapter 7.2, Problem 12E

a.

To determine

Find the critical value $t_{\frac{\alpha }{2}}$ to construct a confidence interval for the given level and sample size.

Answer to Problem 12E

The critical value $t_{\frac{\alpha }{2}}$ to construct a confidence interval for the level 95% and sample size 15 is 2.015.

Explanation of Solution

Calculation:

The given information is that, the level is 90% and the sample size is 6.

Conditions:

• When the required degrees of freedom is less than 200 and not listed in the degrees of freedom column of the Table A.3, then choose the next smaller number in the table.
• When the required degrees of freedom is, greater than 200, then use the z-value in the bottom row of Table A.3 otherwise use Table A.2.

Here the significance level is 0.10 and the degrees of freedom is 5 $\left(=6-1\right)$.

For two-tailed test, the $t_{\frac{\alpha }{2}}$ is

$\begin{array}{c}{t}_{\frac{\alpha }{2}}=t_{\frac{0.10}{2}}\\ =t_{0.05}\end{array}$

Critical value for $t_{0.05}$:

From Table A.3: Critical Values for the Student’s t Distribution

• Locate the value 5 in the first column of the table.
• Locate the value 0.05 in the first row of the table.
• The intersecting value of row and column is 2.015.

Thus, the critical value $t_{\frac{\alpha }{2}}$ to construct a confidence interval for the level 90% and sample size 6 is 2.015.

b.

To determine

Find the critical value $t_{\frac{\alpha }{2}}$ to construct a confidence interval for the given level and sample size.

Answer to Problem 12E

The critical value $t_{\frac{\alpha }{2}}$ to construct a confidence interval for the level 90% and sample size 3 is 2.718.

Explanation of Solution

Calculation:

The given information is that, the level is 98% and the sample size is 12.

Here, the significance level is 0.02 and the degrees of freedom is 11 $\left(=12-1\right)$.

For two-tailed test, the $t_{\frac{\alpha }{2}}$ is

$\begin{array}{c}{t}_{\frac{\alpha }{2}}=t_{\frac{0.02}{2}}\\ =t_{0.01}\end{array}$

Critical value for $t_{0.01}$:

From Table A.3: Critical Values for the Student’s t Distribution

• Locate the value 11 in the first column of the table.
• Locate the value 0.01 in the first row of the table.
• The intersecting value of row and column is 2.718.

Thus, the critical value $t_{\frac{\alpha }{2}}$ to construct a confidence interval for the level 98% and sample size 12 is 2.718.

c.

To determine

Find the critical value $t_{\frac{\alpha }{2}}$ to construct a confidence interval for the given level and sample size.

Answer to Problem 12E

The critical value $t_{\frac{\alpha }{2}}$ to construct a confidence interval for the level 95%and sample size 32 is 2.040.

Explanation of Solution

Calculation:

The given information is that, the level is 95% and the sample size is 32.

The significance level is 0.05 and the degrees of freedom is 31 $\left(=32-1\right)$.

For two-tailed test, the $t_{\frac{\alpha }{2}}$ is

$\begin{array}{c}{t}_{\frac{\alpha }{2}}=t_{\frac{0.05}{2}}\\ =t_{0.025}\end{array}$

Critical value for $t_{0.025}$:

From Table A.3: Critical Values for the Student’s t Distribution

• Locate the value 31 in the first column of the table.
• Locate the value 0.025 in the first row of the table.
• The intersecting value of row and column is 2.040.

Thus, the critical value $t_{\frac{\alpha }{2}}$ to construct a confidence interval for the level 95%and sample size 32 is 2.040.

d.

To determine

Find the critical value $t_{\frac{\alpha }{2}}$ to construct a confidence interval for the given level and sample size.

Answer to Problem 12E

The critical value $t_{\frac{\alpha }{2}}$ to construct a confidence interval for the level 99% and sample size 10 is 3.250.

Explanation of Solution

Calculation:

The given information is that, the level is 99% and the sample size is 10.

Here the significance level is 0.01 and the degrees of freedom is 9 $\left(=10-1\right)$.

For two-tailed test, the $t_{\frac{\alpha }{2}}$ is

$\begin{array}{c}{t}_{\frac{\alpha }{2}}=t_{\frac{0.01}{2}}\\ =t_{0.005}\end{array}$

Critical value for $t_{0.005}$:

From Table A.3: Critical Values for the Student’s t Distribution

• Locate the value 9 in the first column of the table.
• Locate the value 0.005 in the first row of the table.
• The intersecting value of row and column is 3.250.

Thus, the critical value $t_{\frac{\alpha }{2}}$ to construct a confidence interval for the level 99% and sample size 10 is 3.250.