Definition:-Many times the size of the sample that is used to make a test of hypothesis about population mean is small that is (n<30), when population standard deviation is unknown, if the population is normally distributed then the appropriate distribution is t test.
This may be the case because we have limited resources and cannot afford to take a large sample or because of nature of the experiment itself.
For example:- To test a new car model of a car for fuel efficiency (miles per gallon), the company may prefer to use a small sample, All cars included in such a test must be sold as used cars. In the case of a small sample, if the population from which the sample is drawn is (approximately) normally distributed and the population standard deviation σ is known, we can still use the normal distribution to make the test of hypothesis about population mean µ. However if the population is (approximately) normally distributed, the population standard deviation σ is not known, and the sample size is small (n< 30), then the normal distribution is replaced by the t test.
Condition under which the t distribution is used to make test of hypothesis about µ: The t test is used to conduct a test of hypothesis about µ if
· The sample size is small (n<30)
· The population from which sample is drawn is approximately normally distributed.
· The population standard deviation σ is unknown.
Test statistic:- The value of the test statistic t for the sample mean x? is computed as
t= (x?- µ)/ (s/√n) Where s= sample standard deviation.