Simulation from a normal (µ = 12, σ = 1) population

Use this diagram to show that the confidence level can be much less than 95% if you simply replace

with


Firstly leave Use sample SD unchecked, click Accumulate and take 100 samples of size n = 30. Note that about about 95% include µ = 12.

Click Use sample SD to see how the CIs would change with σ replaced by s. Since n is large, the CIs do not change much.

Change the sample size to n = 5 and repeat. The CIs change more when the sample SD is used — their widths vary from sample to sample. Intervals based on s now include µ = 12 much less often than 95% of the time.

Repeat with n = 3.