Confidence interval for a mean

If the population standard deviation, σ, is a known value, a confidence interval for a population mean, µ has the form

The value k arises from the standard normal distribution,

The value k = 1.96 gives a confidence level of 95%, but different confidence levels can be found with other values of k.

k Confidence level
1 0.683
2 0.954
3 0.997
1.645 0.90
1.960 0.95
2.576 0.99

Although 95% confidence intervals are most commonly reported, sometimes k is chosen to give a 90% or 99% confidence interval.

Estimating a probability with different confidence levels

A 95% confidence interval for a probability, π, has the form

Replacing the constant 2 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 standard deviation)

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.)