As long as is known the confidence interval estimate of population mean can be obtained by the method discussed earlier, provided the sample is large. Even if is not known we could replace it by its unbiased estimate , defined by

The pervious 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 pervious tutorials.

Where

The method of construction of a confidence interval is same as for the large samples, with

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 an specific example, the 90% confidence limits may be stated as

.

Where

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.

__Example__: A random sample of 19 students of MBA made an average score of 60 with a standard deviation score of 15. Construct a 95% confidence interval for the mean of the entire MBA class.

__Solution__: 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.

__Lower Limit__

__Upper Limit__

The true mean score

, therefore, lies between 52 and 68 with a 98% confidence.

### Comments

comments