Estimating a population mean
We first examine how to estimate the mean, µ, of a population when the population standard deviation, σ, is a known value. (In practice, σ is usually unknown, but we leave this until later in this section.)
The sample mean, , is approximately normal, with
![]() |
= μ |
![]() |
= | ![]() |
When is used to estimate µ, the error is approximately
error = ![]() |
![]() |
) |
so the standard error of is
.
95% bounds for the error
Applying the 70-95-100 rule of thumb to the error distribution,
Prob( error is between ± 2 | ![]() |
) is approximately 0.95 |
This can be refined using the properties of the normal distribution to get an exact probability of 0.95.
Prob( error is between ± 1.96 | ![]() |
) = 0.95 |
95% confidence interval
Since will be within 1.96
of µ
with probability 0.95, we are 95% confident that µ
is in the interval
This is a 95% confidence interval for µ and the interval has a confidence level of 0.95.
Example
Consider a type of measurement that is normally distributed with known σ but unknown mean, µ:
X ~ normal (μ , σ = 0.0068)
The mean of a random sample of n = 16 values will therefore be normally distributed with standard error
![]() |
= 0.0068 / 4 = 0.0017 |
From this, we can obtain bounds on the error:
P(-0.00333 < error < 0.00333) = 0.95
If the sample data are:
then a 95% confidence interval for µ would be
0.74362 ± 0.00333 = 0.74029 to 0.74695
We are 95% confident that µ is between 0.74029 and 0.74695.