91Ó°ÊÓ

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

Here, df = 5

Consider, at 90% confidence level,

$$\begin{gathered} \mathrm{Î\pm }=1-90\% \\ =1-0.90 \\ =0.10 \end{gathered}$$

Therefore,

$$\begin{gathered} \frac{\mathrm{Î\pm }}{2}=\frac{0.10}{2} \\ =0.05 \end{gathered}$$

Hence, from ${\mathrm{χ}}^2$the table,

$$\begin{gathered} {\mathrm{χ}}_{\frac{\mathrm{Î\pm }}{2}}^2={\mathrm{χ}}_{0.05}^2 \\ =11.0705 \end{gathered}$$

$$\begin{gathered} {\mathrm{χ}}_{1-\frac{\mathrm{Î\pm }}{2}}^2={\mathrm{χ}}_{1-0.05}^2 \\ ={\mathrm{χ}}_{0.95}^2 \\ =1.1455 \end{gathered}$$

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

Here, df = 13

Consider, at 95% confidence level,

$$\begin{gathered} \mathrm{Î\pm }=1-95\% \\ =1-0.95 \\ =0.05 \end{gathered}$$

Therefore,

$$\begin{gathered} \frac{\mathrm{Î\pm }}{2}=\frac{0.05}{2} \\ =0.025 \end{gathered}$$

Hence, from ${\mathrm{χ}}^2$table,

$$\begin{gathered} {\mathrm{χ}}_{\frac{\mathrm{Î\pm }}{2}}^2={\mathrm{χ}}_{0.025}^2 \\ =24.7356 \end{gathered}$$

$$\begin{gathered} {\mathrm{χ}}_{1-\frac{\mathrm{Î\pm }}{2}}^2={\mathrm{χ}}_{1-0.025}^2 \\ ={\mathrm{χ}}_{0.975}^2 \\ =5.0087 \end{gathered}$$

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

Here, df = 28

Consider, at 95% confidence level,

$$\begin{gathered} \mathrm{Î\pm }=1-95\% \\ =1-0.95 \\ =0.05 \end{gathered}$$

Therefore,

$$\begin{gathered} \frac{\mathrm{Î\pm }}{2}=\frac{0.05}{2} \\ =0.025 \end{gathered}$$

$$\begin{gathered} {\mathrm{χ}}_{\frac{\mathrm{Î\pm }}{2}}^2={\mathrm{χ}}_{0.025}^2 \\ =44.4608 \end{gathered}$$

$$\begin{gathered} {\mathrm{χ}}_{1-\frac{\mathrm{Î\pm }}{2}}^2={\mathrm{χ}}_{1-0.025}^2 \\ ={\mathrm{χ}}_{0.975}^2 \\ =38.0477 \end{gathered}$$

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

Here, df = 13

Consider, at 99% confidence level,

$$\begin{gathered} \mathrm{Î\pm }=1-99\% \\ =1 \\ =0.01 \end{gathered}$$

$$\begin{gathered} \frac{\mathrm{Î\pm }}{2}=\frac{0.01}{2} \\ =0.005 \end{gathered}$$

Hence, from X² table,

$$\begin{gathered} {\mathrm{χ}}_{\frac{\mathrm{Î\pm }}{2}}^2={\mathrm{χ}}_{0.005}^2 \\ =29.8195 \end{gathered}$$

$$\begin{gathered} {\mathrm{χ}}_{1-\frac{\mathrm{Î\pm }}{2}}^2={\mathrm{χ}}_{1-0.005}^2 \\ ={\mathrm{χ}}_{0.995}^2 \\ =3.5650 \end{gathered}$$

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