Confidence interval for a proportion
For a sample proportion p-hat from n trials, the standard error is the square root of p-hat(1 - p-hat) / n. A 95 percent interval is p-hat plus or minus 1.96 x SE. It needs enough successes and failures, at least 10 of each.
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Analogy
A poll result comes with a plus-or-minus figure: that is this margin.
Example
p-hat = 0.56, n = 400SE = square root of (0.56 x 0.44 / 400)= 0.0248Margin = 1.96 x 0.0248 = 0.0486Interval: 0.511 to 0.609
In practice
Used in election polls, defect rates and survey results.
Practice
Flashcards
- Standard error of a sample proportion?Square root of p-hat(1 - p-hat) / n.
Interview questions
- A poll of 400 voters finds 56% support a law. Calculate and interpret a 95% confidence interval.
- SE = 0.0248
- Margin = 1.96 x 0.0248 = 0.0486
- Interval 51.1% to 60.9%
- Repeated polls would capture the true share about 95% of the time
Questions people ask
What is the margin of error in a poll?
It is the plus-or-minus part of a confidence interval, half its width. A poll of 56 percent with a 4.9 point margin of error means the true share is plausibly between 51.1 and 60.9 percent. Larger samples give smaller margins.
How do I find the sample size needed for a given margin of error?
For a proportion, use n = z squared x p(1-p) / E squared, with p = 0.5 if unknown, as it gives the largest n. For a margin of 0.03 at 95 percent, n = 3.8416 x 0.25 / 0.0009 = 1067.1, so round up to 1,068.