Chapter 7.2, Problem 1CYU

a.

To determine

Find the critical value $t_{\frac{\alpha }{2}}$ required to construct a confidence interval for the given level and sample size using Table A.3.

Answer to Problem 1CYU

The critical value $t_{\frac{\alpha }{2}}$ required to construct a confidence interval for the level 95% and sample size 15 is 2.145.

Explanation of Solution

Calculation:

The given information is that, the level is 95% and the sample size is 15.

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 0.05 $\left(=1-0.95\right)$ and the degrees of freedom is 14 $\left(=15-1\right)$.

For two-tailed test,

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

Critical value for $t_{0.025}$:

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

• Locate the value 14 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 2.145.

Thus, the critical value $t_{\frac{\alpha }{2}}$ required to construct a confidence interval for the level 95% and sample size 15 is, 2.145.

b.

To determine

Find the critical value $t_{\frac{\alpha }{2}}$ required to construct a confidence interval for the given level and sample size using Table A.3.

Answer to Problem 1CYU

The critical value $t_{\frac{\alpha }{2}}$ required to construct a confidence interval for the level 99% and sample size 22 is 2.831.

Explanation of Solution

Calculation:

The given information is that, the level is 99% and the sample size is 22.

Here the significance level is 0.01 and the degrees of freedom is 21 $\left(=22-1\right)$.

For two-tailed test,

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

Critical value for $t_{0.005}$:

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

• Locate the value 21 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 2.831.

Thus, the critical value $t_{\frac{\alpha }{2}}$ required to construct a confidence interval for the level 99% and sample size 22 is 2.831.

c.

To determine

Find the critical value $t_{\frac{\alpha }{2}}$ required to construct a confidence interval for the given level and sample size using Table A.3.

Answer to Problem 1CYU

The critical value $t_{\frac{\alpha }{2}}$ required to construct a confidence interval for the level 90%and sample size 63 is 1.671.

Explanation of Solution

Calculation:

The given information is that, the level is 90% and the sample size is 63.

The significance level is 0.10 and the degrees of freedom is 62 $\left(=63-1\right)$.

Here, the degrees of freedom is not listed in the Table A.3. Hence, choose previous row with degrees of freedom smaller than 62. That is, the previous row with smaller degrees of freedom would be 60.

For two-tailed test,

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

Critical value for $t_{0.05}$:

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

• Locate the value 60 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 1.671.

Thus, the critical value $t_{\frac{\alpha }{2}}$ required to construct a confidence interval for the level 90%and sample size 63 is, 1.671.

d.

To determine

Find the critical value $t_{\frac{\alpha }{2}}$ required to construct a confidence interval for the given level and sample size using Table A.3.

Answer to Problem 1CYU

The critical value $t_{\frac{\alpha }{2}}$ required to construct a confidence interval for the level 95% and sample size 2 is, 12.706.

Explanation of Solution

Calculation:

The given information is that, the level is 95% and the sample size is 2.

Here, the significance level is 0.05 and the degrees of freedom is 1 $\left(=2-1\right)$.

For two-tailed test,

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

Critical value for $t_{0.025}$:

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

• Locate the value 1 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 12.706.

Thus, the critical value $t_{\frac{\alpha }{2}}$ required to construct a confidence interval for the level 95% and sample size 2 is 12.706.