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, we can therefore write

Prob( error is between ± 2 )  ≈  0.95

The value 1.96 could be used instead of 2 in this equation since exactly 95% of values from a normal distribution are within 1.96 standard deviations of the mean. However this refinement does not help because:


Management succession plans

An earlier example described whether a sample of 210 chief executives of fast-growing small companies had management succession plans to deal with the consequences of executives resigning.

Management succession plan?     Frequency
Yes 107
No 103
Total     210

The proportion of companies with management succession plans was p = 107/210 = 0.510 and this provides a point estimate of the probability, π, that other similar companies will have such plans.

The diagram below shows our estimated normal distribution for the errors and approximate 95% bounds on the error.

95% confidence interval

Since there is a probability of approximately 0.95 that p is within

2 ×

of π, a 95% confidence interval for π is...

A 95% confidence interval for the population proportion of small fast-growing companies with management succession plans is therefore

0.510 ± 0.069

i.e. 0.441  to   0.579

We are therefore 95% confident that between 44% and 58% of small fast-growing companies have management succession plans.