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.