Question

For each of the following combinations of confidence interval and degrees of freedom (df), use Table IV in Appendix D to find the values of ${\chi }_{\frac{\alpha }{2}}^2$and ${\chi }_{1-\frac{\alpha }{2}}^2$.

a. 90% confidence interval with df = 5

b. 95% confidence interval with df = 13

c. 95% confidence interval with df = 28

d. 99% confidence interval with df = 13

Short answer

a. The values of ${\chi }_{\frac{\alpha }{2}}^2$and ${\chi }_{1-\frac{\alpha }{2}}^2$for 5 degrees of freedom at a 90% confidence level are 11.0705 and 1.1455.

b.The values of ${\chi }_{\frac{\alpha }{2}}^2$and ${\chi }_{1-\frac{\alpha }{2}}^2$for 13 degrees of freedom at 95% confidence level are 24.7356 and 5.0087.

c.The values of ${\chi }_{\frac{\alpha }{2}}^2$and ${\chi }_{1-\frac{\alpha }{2}}^2$for 28 degrees of freedom at 95% confidence level are 44.4608 and 38.0477.

d. The values of ${\chi }_{\frac{\alpha }{2}}^2$and ${\chi }_{1-\frac{\alpha }{2}}^2$for 13 degrees of freedom at a 99% confidence level are 29.8195 and 3.5650.

Step-by-step solution

Given information

For each part, the values of degrees of freedom and confidence level are given.

(a) Finding the values for  χα22 and χ1-α22 for 5 degrees of freedom

Here, df = 5

Consider, at 90% confidence level,

$$\begin{gathered} \alpha =1-90\% \\ =1-0.90 \\ =0.10 \end{gathered}$$

Therefore,

$$\begin{gathered} \frac{\alpha }{2}=\frac{0.10}{2} \\ =0.05 \end{gathered}$$

Hence, from ${\chi }^2$the table,

$$\begin{gathered} {\chi }_{\frac{\alpha }{2}}^2={\chi }_{0.05}^2 \\ =11.0705 \end{gathered}$$

$$\begin{gathered} {\chi }_{1-\frac{\alpha }{2}}^2={\chi }_{1-0.05}^2 \\ ={\chi }_{0.95}^2 \\ =1.1455 \end{gathered}$$

Thus, the values of ${\chi }_{\frac{\alpha }{2}}^2$and ${\chi }_{1-\frac{\alpha }{2}}^2$for 5 degrees of freedom at 90% confidence level are 11.0705 and 1.1455.

(b) Finding the values for χα22and χ1-α22 for 13 degrees of freedom

Here, df = 13

Consider, at 95% confidence level,

$$\begin{gathered} \alpha =1-95\% \\ =1-0.95 \\ =0.05 \end{gathered}$$

Therefore,

$$\begin{gathered} \frac{\alpha }{2}=\frac{0.05}{2} \\ =0.025 \end{gathered}$$

Hence, from ${\chi }^2$table,

$$\begin{gathered} {\chi }_{\frac{\alpha }{2}}^2={\chi }_{0.025}^2 \\ =24.7356 \end{gathered}$$

$$\begin{gathered} {\chi }_{1-\frac{\alpha }{2}}^2={\chi }_{1-0.025}^2 \\ ={\chi }_{0.975}^2 \\ =5.0087 \end{gathered}$$

Thus, the values of ${\chi }_{\frac{\alpha }{2}}^2$and ${\chi }_{1-\frac{\alpha }{2}}^2$for 13 degrees of freedom at 95% confidence level are 24.7356 and 5.0087.

(c) Finding the values for χα22and χ1-α22 for 28 degrees of freedom

Here, df = 28

Consider, at 95% confidence level,

$$\begin{gathered} \alpha =1-95\% \\ =1-0.95 \\ =0.05 \end{gathered}$$

Therefore,

$$\begin{gathered} \frac{\alpha }{2}=\frac{0.05}{2} \\ =0.025 \end{gathered}$$

Hence, from X² table,

$$\begin{gathered} {\chi }_{\frac{\alpha }{2}}^2={\chi }_{0.025}^2 \\ =44.4608 \end{gathered}$$

$$\begin{gathered} {\chi }_{1-\frac{\alpha }{2}}^2={\chi }_{1-0.025}^2 \\ ={\chi }_{0.975}^2 \\ =38.0477 \end{gathered}$$

Thus, the values of ${\chi }_{\frac{\alpha }{2}}^2$and ${\chi }_{1-\frac{\alpha }{2}}^2$for 28 degrees of freedom at 95% confidence level are 44.4608 and 38.0477.

(d) Finding the values for χα22and χ1-α22for 13 degrees of freedom

Here, df = 13

Consider, at 99% confidence level,

$$\begin{gathered} \alpha =1-99\% \\ =1 \\ =0.01 \end{gathered}$$

Therefore,

$$\begin{gathered} \frac{\alpha }{2}=\frac{0.01}{2} \\ =0.005 \end{gathered}$$

Hence, from X² table,

$$\begin{gathered} {\chi }_{\frac{\alpha }{2}}^2={\chi }_{0.005}^2 \\ =29.8195 \end{gathered}$$

$$\begin{gathered} {\chi }_{1-\frac{\alpha }{2}}^2={\chi }_{1-0.005}^2 \\ ={\chi }_{0.995}^2 \\ =3.5650 \end{gathered}$$

Thus, the values of ${\chi }_{\frac{\alpha }{2}}^2$and ${\chi }_{1-\frac{\alpha }{2}}^2$for 13 degrees of freedom at 99% confidence level are 29.8195 and 3.5650.