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?
The central limit theorem and estimator in the task
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statistics
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2You know the sample mean, so all you need is the sample standard deviation. How do you get it? – 2017-01-29
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0How can I get standard deviation? – 2017-01-29
1 Answers
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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 $$ \left[\hat p \pm Z_{0.975}\left(\frac{\hat p (1 - \hat p) }{n}\right)^{1/2}\right] $$