Confidence interval coverage
Higher confidence produces wider intervals
Repeat a study of 20 observations from a normal population with fixed mean 0 and SD 1. Each sample produces a different t interval.
Keep the generated samples identical and raise confidence from 95% to 99%. Compare the interval widths and coverage.
Repeated intervals around a fixed μ
Each horizontal segment is a confidence interval from a different sample. The vertical reference line is the fixed population mean. Dashed, triangle-marked intervals miss that mean.
Widths of the simulated intervals
These bars count the widths of the simulated intervals. Wider intervals express less precise estimates.
Worked comparison
Follow the calculation
What the symbols mean
x̄ is the sample mean and SE is its standard error. The t interval uses the sample standard deviation s and a t critical value with n − 1 degrees of freedom. The z interval uses the known population standard deviation σ and a normal critical value.
Assumptions and full calculation
The applet draws independent samples from a normal population. Each t interval uses its sample's standard deviation and n − 1 degrees of freedom. Each z interval uses the known population standard deviation. The population mean stays fixed. The plot shows the first 80 intervals. The coverage and mean width calculations use every simulated interval. Solid intervals with circles contain μ. Dashed intervals with triangles miss it. The confidence level describes the procedure over repeated studies. It does not give a probability that the fixed mean lies inside one calculated interval.