Any statistical test that uses t distribution can be called a t-test, or the "student's t-test". It is basically used when the sample size is small i.e. n<30.
For example, if a person wants to test the hypothesis that mean height of student's of a college is not different from 150 cms, he can take a sample of size say 20 from the college. From the mean height of these students, he can test the hypothesis. The test to be used for this purpose is t-test.
There are different types of t-test each for different purpose. Some of the popular types are outlined below:
The example on heights of students explained above requires this test. In that example, μ0 = 150.
For example suppose one is interested to test if there is any significant difference between the mean height of male and female students in a particular college. In such a situation, t-test for difference of means can be applied. One would have to take two independent samples from the college- one from males and the other from females in order to perform this test.
An additional assumption of this test is that the variance of the two populations is equal.
For example we may wish to test if a newly developed intervention program for disadvantaged students is useful. For this, we need to obtain scores from say 22 students in a standardized test before administering the program. After the program is over, the same test needs to be administered again on the same group of 22 students and scores obtained.
The two samples- the sample of prior intervention scores and the sample of post intervention scores are related as each student has two scores. The samples are therefore dependent. The paired t-test can is applicable in such scenarios.
For example suppose a random sample of 27 pairs of observation from a normal population gave a correlation coefficient 0.2. Notice that this is the sample correlation coefficient obtained from a sample of observations. One may be interested to know whether the variables are correlated in the population. In this case we can use t-test for correlation coefficient
Irrespective of the type of t-test used, two assumptions have to be met.
Student's t-test is a small sample test. It is difficult to drawn a clearest line of demarcation between large and small sample. Statisticians have generally agreed that a sample may be considered small if its size is < 30.
The test used for dealing with problems relating the large samples are different from the one used for small samples. We often use z-test for large samples.