A confidence interval gives a plausible range for a population parameter based on sample data and a method with known long-run coverage. A 95% interval means the method captures the true parameter in about 95% of repeated samples, not that this one interval has 95% probability.
- Confidence level describes method reliability across repeated samples.
- Margin of error depends on variability, sample size, and critical value.
- A narrow interval is not useful if assumptions are violated.
General interval form
Most intervals follow estimate +/- critical value * standard error. The estimate could be a sample mean xbar or sample proportion phat.
Use z critical values when conditions justify normal approximation and population standard deviation is known (or large-sample approximation for proportions). Use t critical values for means when sigma is unknown and sampled from a roughly normal population or sufficiently large n.
Interpretation and planning sample size
Correct interpretation references repeated sampling performance of the method. It does not assign probability to a fixed unknown parameter after data collection.
To reduce margin of error, increase sample size. Because error scales with 1/sqrt(n), shrinking error substantially can require much larger n.
- State parameter, level, and units in one sentence.
- Check conditions before reporting the interval.
- Keep more precision in intermediate calculations than in final display.
Build a 95% interval for a mean
A sample of n=64 task completion times has mean 42 minutes. Assume population standard deviation sigma=16 minutes.
- Use z interval: xbar +/- z* * sigma/sqrt(n), with z* = 1.96 for 95% confidence.
- Compute standard error: 16/sqrt(64)=2.
- Compute margin of error: 1.96*2=3.92.
- Form interval: 42 +/- 3.92, giving (38.08, 45.92).
Common mistakes
- Saying there is a 95% chance the fixed parameter lies in this computed interval.
- Using z when a small-sample t interval is needed.
- Reporting an interval without checking assumptions.
- Rounding too early and distorting margin of error.
Try one
What happens to margin of error if sample size quadruples, all else equal?
It is cut in half, because margin of error scales with 1/sqrt(n).
Sources
- OpenStax: Introductory StatisticsOpen textbook covering descriptive statistics, probability, inference, and regression fundamentals.
- Khan Academy: Statistics and probabilityClear instructional coverage of core statistics ideas with worked examples and practice.