Point and interval estimates
In this section, we show how to use an interval estimate to indicate the
accuracy of the sample mean, ,
as an estimate of a population mean, µ.
Pages printed by inkjet printer
The manufacturer of a new line of inkjet printers conducts an experiment to estimate the mean number of pages that users can expect to print with each print cartridge. The dot plot below shows the number of pages printed by a sample of 10 cartridges.
We could report that
However the underlying population mean pages printed, µ, is unlikely to be exactly this. It is more informative to make a statement such as:
(The details will be explained later.)
Error distribution
The error distribution is the key to interval estimation. For most common parameter estimates, we can find the error distribution (or an approximation to it).
Error distribution for mean
To estimate the mean, µ,
of a population whose standard deviation is σ,
we would use the mean of a random sample of n values, .
The resulting estimation error has approximately a normal distribution,
error ~ normal (0, | ![]() |
) |
(This distribution is exact if the population is normal and close if the sample size, n, is large.)
If the population standard deviation, σ, is unknown, we can approximate this distribution,
error ~ normal (0, | ![]() |
) |
95% bounds on the error
From the error distribution, we can find a range of values within which the error will lie with probability 0.95,
Expressed in an equation,
Prob ( -e* < error < e* ) = 0.95
95% confidence interval
If the estimation error is between -e* and e*, then the estimator is within e* of the parameter that we are estimating, so
Prob ( estimate - e* < parameter < estimate + e* ) = 0.95
We therefore call the interval
estimate - e* to   + e*
a 95% confidence interval and we have 95% confidence that it will include the unknown parameter value.
Confidence interval from standard error
The 70-95-100 rule of thumb states that about 95% of values in most distributions are within 2 standard deviations of the mean. For unbiased estimators (with zero mean), we therefore have the approximation:
In terms of the standard error, a 95% confidence interval can therefore be found as
estimate - 2 s.e. to estimate + 2 s.e.
The standard error of most commonly used estimators can be readily found by either a formula or statistical software, so this result is extremely useful in practice.
If you know the standard error of an estimator, you can find an approximate 95% confidence interval.
Refinements
The confidence level of a confidence interval that is found in this way is 95% only if we know the exact error distribution. Since we must often approximate the error distribution (e.g. by replacing σ by s when estimating a population mean), the confidence level may be only approximately 95%.
For some estimators, we will give a refinement to this type of confidence interval to make the confidence level closer to 95%. However the '± 2 s.e.' approximation is a useful guide in most circumstances.