What is the confidence level of each of the following confidence intervals $\mathbf{\mu }$?

1. $\overline{\mathbf{\chi }}\mathbf{\pm }\mathbf{1}\mathbf{.}\mathbf{96}\left(\frac{\mathbf{\sigma }}{\sqrt{\mathbf{n}}}\right)$
2. $\overline{\mathbf{\chi }}\mathbf{\pm }\mathbf{1}\mathbf{.}\mathbf{645}\left(\frac{\mathbf{\sigma }}{\sqrt{\mathbf{n}}}\right)$
   
3. $\overline{\mathbf{\chi }}\mathbf{\pm }\mathbf{2}\mathbf{,}\mathbf{575}\left(\frac{\mathbf{\sigma }}{\sqrt{\mathbf{n}}}\right)$
4. $\overline{\mathbf{\chi }}\mathbf{\pm }\mathbf{1}\mathbf{.}\mathbf{282}\left(\frac{\mathbf{\sigma }}{\sqrt{\mathbf{n}}}\right)$
5. $\overline{\mathbf{\chi }}\mathbf{\pm }\mathbf{.}\mathbf{99}\left(\frac{\mathbf{\sigma }}{\sqrt{\mathbf{n}}}\right)$
   

The formula to calculate the level of confidence, the confidence interval for the mean is:

$\overline{X}{\text{±z}}_{\text{1}}\text{=}\alpha \text{/}2\left(\frac{\sigma }{\sqrt{\text{n}}}\right)$

We will calculate the level of confidence for each provided confidence interval using the score of the multiplied or the important test statistic.

The calculation is given below:

$$\begin{gathered} {\text{z}}_{\text{1}}=\alpha /2=1.96 \\ \text{1}-\alpha /2=0.975 \\ \alpha =0.05 \end{gathered}$$

Therefore,

$$\begin{gathered} \left(1-\alpha \right)\times 100\% \\ =95\% \end{gathered}$$

Hence, the level of confidence is 95%

$$\begin{gathered} {\text{z}}_{\text{1}}=\alpha /2=1.645 \\ 1-\alpha /2=0.95 \\ \alpha =10 \end{gathered}$$

Therefore,

$$\begin{gathered} \left(1-\alpha \right)\times 100\% \\ =90\% \end{gathered}$$

Hence, the level of confidence is 90%

$$\begin{gathered} {\text{z}}_{\text{1}}=\alpha /2=2.575 \\ \text{1}-\alpha /2=0.995 \\ \alpha =0.01 \end{gathered}$$

Therefore,

$$\begin{gathered} \left(1-\alpha \right)\times 100\% \\ =99\% \end{gathered}$$

Hence, the level of confidence is 99%

Step 3: (d) The data is given below

$$\begin{gathered} {\text{Here, test stastticsZ}}_{\text{cr}}\text{=1.282} \\ \text{Confidence level=80\%} \end{gathered}$$

Hence, the level of confidence is 80%

Step 3: (e) The data is given below

$$\begin{gathered} {\text{z}}_{\text{1}}\text{=}\alpha \text{/}2=0.99 \\ 1-\alpha \text{/}2=0.839 \\ \alpha =0.322 \end{gathered}$$

Therefore,

$$\begin{gathered} \left(1-\alpha \right)\times 100\% \\ =67.8\% \end{gathered}$$

Hence, the level of confidence is 67.8%