Confidence interval for a mean (known σ)
Replacing the constant 1.96 with other values alters the confidence level of the confidence interval for a population mean.
Simulation
We now demonstrate how altering the constant changes the confidence level in a simulation.
Estimating a probability with different confidence levels
A 95% confidence interval for a probability, π, has the form
The constant 2 in this formula again arose from the standard normal distribution. Replacing it with 1.645 gives an interval with approximately a 90% confidence level, and using 2.576 results in a 99% confidence level.
Estimating a population mean (unknown σ)
When the population standard deviation, σ, is unknown, a 95% confidence interval for µ has the form
where tn-1 is obtained from a table. Changing the confidence level to 90% or 99% involves changing this constant. The appropriate value can again be obtained from a table. (We give no further details here.)