If samples of sizeare taken from a population whose meanand standard deviationis known thenthe mean of the sample has the normal distributionIf we know the standard deviation but not the mean of the population then we can find a confidence interval for the mean of the population by rearranging(1) to give (2) with probability corresponding to the value of z. Hence the confidence interval for the mean is given by(3)
Note that confidence intervals are two sided. If we are required to find a 90% confidence interval the we look up that value of z corresponding to a probability ofin the tables for the normal distribution.
In practice, the standard deviation is only one more thing to be calculated from the data, so there is rarely such a thing as the ‘true’ standard deviationIn the case where the population is normal but the standard deviation has to be calculated from the sample we cannot use the above expression for the confidence interval. Instead we use student’s– distribution. The– distribution is similar to the normal distribution, being symmetrical, bell shaped and having most most values occurring within three or so standard deviations from the mean. In addition asthe– distribution approximates more closely to the normal distribution.
Ifis the standard deviation calculated from the sample of sizethen instead of (1) we have and instead of (2) we have and (3) becomes
Example: Find a 95 %confidence interval for the mean of the population from which the following sample is taken, assuming that the population is normally distributed.
from the tables.
The confidence interval is