Standard error of an ad click-through rate

Question

In a sample of n = 400 users, a fraction \hat{p} = 0.30 clicked an ad. What is the standard error of the sample proportion?

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A proportion is just an average of 0s and 1s (did not click / clicked), so it has a standard error too. For a proportion, the spread of a single yes/no outcome is built right into p, and the standard error is 	ext{SE} = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}. Plug in \hat{p} = 0.30 and n = 400: 	ext{SE} = \sqrt{\frac{0.30 	imes 0.70}{400}} = \sqrt{\frac{0.21}{400}} = \sqrt{0.000525} \approx 0.0229. So the measured 30\% click rate carries a standard error of about 2.3 percentage points: rerun the same test and you would expect the click rate to land somewhere around 30\%, give or take a couple…

Standard error of an ad click-through rate

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