Given the following parameters
[tex]\begin{gathered} \sigma\Rightarrow s\tan dard\text{ deviation=0.4} \\ n\Rightarrow\text{sample size=20} \\ \text{Significance level}\Rightarrow95\text{ \%} \\ z_{\frac{\alpha}{2}}=1.960 \end{gathered}[/tex]
To find the mean of the data, we will have to use
[tex]\begin{gathered} \bar{x}=\frac{\Sigma x_i}{n} \\ \Sigma x_i=3497.76 \\ n=20 \\ \bar{x}=\frac{3497.76}{20} \\ =174.888 \end{gathered}[/tex]
Using the confidence interval formula of
[tex]CI=\bar{x}\pm z_{\frac{\alpha}{2}}\times\frac{\sigma}{\sqrt[]{n}}[/tex]
Substitute for all values to find the confidence interval.
[tex]\begin{gathered} CI=174.888\pm1.960\times\frac{0.4}{\sqrt[]{20}} \\ =174.888\pm1.960\times0.0894427191 \\ =174.888\pm0.175 \end{gathered}[/tex]
Hence, the confidence interval is
[tex]174.888\pm0.175[/tex]
The critical value is
[tex]1.960[/tex]
The standard error of the mean is
[tex]\sigma_{\bar{x}}=\frac{\sigma}{\sqrt[]{n}}=\frac{0.4}{\sqrt[]{20}}=0.089[/tex]
The confidence interval is
[tex](174.71,175.07)[/tex]
