Confidence interval for a mean (known standard deviation)

At the start of this chapter, we developed a 95% confidence interval for a population mean, µ, when the standard deviation, σ, was a known value. This was of the form

We showed that intervals of this form have probability 0.95 of containing the population mean, µ.

Changing the constant 1.96

We next generalise to consider interval estimates of the form

where k can take values other than 1.96.

When k becomes smaller, the intervals become narrower so they have a lower probability of including µ — in other words, the confidence level becomes lower. The relationship between k and the confidence level is determined by the standard normal distribution,

The table below gives some examples of k and the resulting confidence level.

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.

Simulation

The diagram below shows a random sample from a normal population with µ = 12 and σ = 2.

Click Accumulate and take about 50 samples. Now use the slider to adjust the constant k.

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 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.)