Show that the inequality on page 304 leads to the following \((1-\alpha) 100 \%\) confidence limits:

\[\frac{x+\frac{1}{2} z_{\alpha / 2}^{2} \pm z_{\alpha / 2} \sqrt{\frac{x(n-x)}{n}+\frac{1}{4} z_{\alpha / 2}^{2}}}{n+z_{\alpha / 2}^{2}}\]

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EXAMPLE 2 Conservative 95% confidence interval for binomial proportion p Refer to Example 1 where 8 out of 55 wind turbines were too noisy according to new restricted guidelines. Using computer software (see Exercise 10.13) obtain the 95% conservative confidence interval for the proportion of all wind turbines managed by the company that are too noisy.