I am studying descriptive statistic and I am trying to do this exercise but I am not getting.
A certain manufacturer guarantees that his machines produce only $5\%$ of defective parts. In order to verify the veracity of this statement, $1000$ pieces produced by a machine were analyzed, of which $40$ pieces had some type of defect. Build a $95\%$ confidence interval for the proportion of defective parts and discuss the manufacturer's claim.
$\hat p=\frac{40}{1000}$, so a 95% confidence interval for the true population proportion is $\hat p\pm z*s.e.=\frac{40}{1000}\pm 1.96 \sqrt{\frac{\frac{40}{1000}\left(1-\frac{40}{1000}\right)}{1000}}=(0.02785432, 0.05214568)$. 5% is contained in this interval, so we believe the manufacturer's claim that 5% of his items are defective is reasonable.