Estimate with uncertainty.
For a sufficiently large random sample, estimate the standard error with . An approximate interval is p̂ ± z*SE, where z* depends on the confidence level. For a 95% interval, z* is approximately 1.96.
The confidence level describes the long-run proportion of intervals constructed by this method that contain the fixed population proportion. It does not say that 95% of individual observations lie within this interval. Greater confidence widens the interval; a larger sample narrows it.
Try it. Watch it change.
Each simulated sample uses independent Bernoulli trials. The population p stays fixed while p̂ varies. Before a draw, p is used as the illustrative estimate. Bounds outside [0, 1] are clipped only in the picture.
Explore: Keep the observed proportion fixed. Increase n, then increase the confidence level, and compare interval widths.
In a random sample of 400 people, 240 support a proposal. Find an approximate 95% interval.
- p̂ = = 0.6.
- SE = ≈ 0.024495.
- 0.6 ± 1.96SE gives approximately (0.5520, 0.6480).
Assumed knowledge
Sampling distributions and standard normal quantiles.
Learning checkpoints & source
YOUR LEARNING CHECKPOINT- Construct an approximate confidence interval.
- Interpret coverage and effects of confidence level and sample size.