Confidence Interval for the Mean of a Normal Distribution When the Standard Deviation is not Given
If samples of sizearetaken from a population whose meanandstandard deviationisknown thenthemean of the sample has the normal distributionIfwe know the standard deviation but not the mean of the populationthen we can find a confidence interval for the mean of the populationby rearranging(1)to give(2) with probability corresponding to the value of z. Hence theconfidence interval for the mean is given by(3)
Note that confidenceintervals are two sided. If we are required to find a 90% confidenceinterval the we look up that value of z corresponding to aprobability ofinthe tables for the normal distribution.
In practice, the standard deviation is only one morething to be calculated from the data, so there is rarely such a thingas the 'true' standard deviationInthe case where the population is normal but the standard deviationhas to be calculated from the sample we cannot use the aboveexpression for the confidence interval. Instead we use student's–distribution. The–distribution is similar to the normal distribution, beingsymmetrical, bell shaped and having most most values occurring withinthree or so standard deviations from the mean. In addition asthe–distribution approximates more closely to the normal distribution.
Ifisthe standard deviation calculated from the sample of sizetheninstead of (1) we have andinstead of (2) we have and (3) becomes
Example: Find a 95 %confidence interval for the meanof the population from which the following sample is taken, assumingthat the population is normally distributed.
The confidence interval is