Answer
$0.831\lt p\lt0.913$
Work Step by Step
We have $\hat p=\frac{157}{180}=0.872, \hat q=1-\hat p=0.128 , n=180 $
At a 90% confidence the critical z-value is $z_{\alpha/2}=1.65 $
The margin of error can be found as
$E=z_{\alpha/2}\times\sqrt {\frac{\hat p\hat q}{n}}=1.65\times\sqrt {\frac{0.872\times0.128}{180}}=0.041$
Thus, the interval of the true proportion can be found as
$\hat p-E\lt p\lt\hat p+E$ which gives $0.831\lt p\lt0.913$
