TITLE: The central limit theorem and estimator in the task
QUESTION [0 upvotes]: I have problem with task:
Among 500 randomly selected articles found 25 faulty copies. Estimate (a confidence level of 0.95) the percentage of defective products throughout the production batch.
How to start solve this?

REPLY [0 votes]: Let $p$ be the real proportion of defects, so you are estimating it with the sample proportion $\hat p$ that in your case equals $25/500$. 
Assuming that the articles are independent identically distributed with $\mathcal B er (p)$ each, thus you can approximate the sample average distribution with the Gaussian r.v., i.e., 
$$
\hat p \xrightarrow{D}N\left(p,\frac{p(1-p)}{n}\right).
$$
As such, the $95\%$ CI will be 
$$
\left[\hat p \pm Z_{0.975}\left(\frac{\hat p (1 - \hat p) }{n}\right)^{1/2}\right]
$$
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