The basic formula for estimating a confidence interval for a large sample is
[tex] ( {\displaystyle {\hat {\theta }}} - z_{\alpha/2} \sigma_{{\displaystyle {\hat {\theta }}}} , {\displaystyle {\hat {\theta }}} + z_{\alpha/2} \sigma_{{\displaystyle {\hat {\theta }}}}) [/tex]
Where [tex] \theta [/tex] is the parameter to be estimated.
[tex] {\displaystyle {\hat {\theta }}} [/tex] is the point estimator of the [tex] \theta [/tex] parameter
[tex] \sigma_{\displaystyle {\hat {\theta }}} [/tex] is the standard error
and [tex] Z_{\alpha/2} [/tex] for this case is 2,575.
We estimate the average population, so:
[tex]\theta = \mu[/tex]
[tex] \overline{x} [/tex] is the estimator of [tex] \mu [/tex]
[tex] \sigma_{\displaystyle {\hat {\theta }}} = \sigma / \sqrt{n} [/tex] is the error.
Then the confidence interval is given by:
[tex] (12.5 - 2.575\frac{4}{\sqrt{40}} , 12.5 + 2.575\frac{4}{\sqrt{40}}) [/tex]
[tex] (10.871 , 14.129) [/tex]
