We have the following information from the question:
[tex]\begin{gathered} sample\text{ size, n=18} \\ \text{standard deviation=0.4} \\ \text{sample mean, }\bar{\text{x}}=\frac{sum\text{ of data values}}{\text{number of data values}} \\ \bar{x}=\frac{3149.94}{18} \\ \bar{x}\cong175 \end{gathered}[/tex]
a) To get the critical value, we would find the significance level first.
Thus, we have:
[tex]\begin{gathered} \text{Significance level, }\alpha=1-confidence\text{ interval} \\ C.I=80\text{\%=0.8} \\ \alpha=1-0.8 \\ \alpha=0.2 \end{gathered}[/tex][tex]\begin{gathered} \text{Critical value=Z}_{\frac{\alpha}{2}}=Z_{\frac{0.2}{2}}=Z_{0.1}=1.28\text{ ( from the z-table)} \\ \text{Therefore, critical value=}\pm\text{1.28}0 \end{gathered}[/tex][tex]\begin{gathered} S\tan dard\text{ deviation of the sample mean, }\sigma_{\bar{x}}=\frac{\sigma}{\sqrt[]{n}} \\ \sigma_{\bar{x}}=\frac{0.4}{\sqrt[]{18}}=0.0943 \end{gathered}[/tex]
b) To find the confidence interval, we have to obtain the Margin of Error first.
[tex]\text{Margin of Error,E}=\text{critical value}\times\text{standard deviation of the sample mean(standard error)}[/tex][tex]\begin{gathered} E=1.28\times0.0943 \\ E=0.1207 \end{gathered}[/tex]
Therefore, the Confidence Interval is:
[tex]\bar{x}-E<\mu<\bar{x}+E[/tex][tex]\begin{gathered} 175-0.1207<\mu<175+0.1207 \\ 174.88<\mu<175.12 \end{gathered}[/tex]
