As long as is known, the confidence interval estimate of a population mean can be obtained by the method discussed earlier, provided the sample is large. Even if is not known we could replace it with its unbiased estimate , defined by
The previous method of estimation fails to provide good estimations if the sample size is small (smaller than 30). In such cases, which are quite frequent, we can use the statistics as defined in previous tutorials.
The method of constructing a confidence interval is the same as for large samples, with
being replaced by . The applications of this statistic, however, presume that the population is approximately bell shaped. A 100% confidence interval estimate for the mean may be obtained as follows:
The 100% confidence limits for the means of the population are, therefore, . As a specific example, the 90% confidence limits may be stated as .
Here is the value of from the distribution table at 5% level of significance corresponding to the given degrees of freedom . For further understanding, let us consider the following practical example.
A random sample of 19 MBA students scored an average of 60 with a standard deviation score of 15. Construct a 95% confidence interval for the mean of the entire MBA class.
Since the sample size is smaller than 30, we will use the statistic to construct the required confidence interval. We are provided with
The degree of freedom
Consulting the distribution table for 15 degrees of freedom, we have
Hence, using formula , the 95% confidence limits would be:
The true mean score , therefore, lies between 52 and 68 with 98% confidence.