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   =    − µ   ~   normal (0,   )

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.