There are various ways we can find out how extreme an event is. We can find the interquartile range and say whether or not the event is an outlier. We can apply the z transform if it is a normal distribution and find the probability of events more extreme. There are more, but all of these methods involve some sort of calculation before an answer can be given. Confidence intervals allow us to just say if an event is extreme with maybe more precision, but they allow more than this. We can find confidence intervals for any statistical parameter, or sample parameter. Most often we want to work out confidence intervals for the true mean and true standard deviation given a sample of a certain size
The meaning of a confidence interval is this: if we have a 95% confidence interval for the sample mean and we take lots of samples, 95 % of the means of those samples will lie within the confidence interval.
If we have a 95% confidence interval for the true mean, there is a 95% probability that the true mean is in the confidence interval. If we find lots of confidence intervals for the true mean, 95% of these confidence intervals will contain the true mean.
We illustrate for a normal distribution. Suppose then that we have a sample of size n. The mean is
– this is the true mean not the mean of the sample and the standard deviation is %sigma – the true standard deviation not the standard deviation of the sample. The confidence interval is
where z is found from the normal distribution tables for each level of confidence. For a 95% confidence interval we want the area of 2.5% of the sample means to be too big and 2.5% to be to small, so using
we look up the corresponding value of z, obtaining 1.96.
Example: It is required to find a confidence interval for the lifetime of bulbs as part of a quality assurance exercise. The lifetimes of the bulbs are known to be normally distributed with mean lifetime 1000 hours and standard deviation 50 hours. Construct a 90% confidence interval for the mean lifetime of samples of 100.
From the normal distribution tables we find a value of z corresponding to
obtaining
The confidence interval is
(991.775, 1007.775)