95% bounds on the estimation error
When sample proportion p is used to estimate a corresponding population proportion, π, the resulting error has the approximate distribution,
error = p − π ~ normal (0, | ![]() |
) |
Replacing π by our best estimate, p , and using the properties of the normal distribution,
Prob( error is between ± 2 | ![]() |
) ≈ 0.95 |
95% confidence interval
A 95% confidence interval for π is therefore...
Example
In a random sample of n = 36 values, there were x = 17 successes. We estimate the population proportion, π, with p = 17/36 = 0.472. The approximate normal distribution for the errors is shown below.
A 95% confidence interval for π is therefore
0.472 ± 0.166
i.e. 0.306 to 0.638
We are therefore 95% confident that the population proportion of successes is between 30.6% and 63.8%. A sample size of n = 36 is clearly too small to give a very accurate estimate.