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.