The dependent t-test for paired samples is used when the samples are paired. This implies that each individual observation of one sample has a unique corresponding member in the other sample.
The emphasis being on pairing of observations, it is obvious that the samples are dependent - hence the name.
Any statistical test involving paired samples and using t-distribution can be called 't-test for paired samples'.
Let us illustrate the meaning of a paired sample. Suppose that we are required to examine if a newly developed intervention program for disadvantaged students has an impact. For this purpose, we need to obtain scores from a sample of n such students in a standardized test before administering the program.
After the program is over, the same test needs to be administered to the same group of students and scores obtained again.
There are two samples: 1) the sample of prior intervention scores (pretest) and, 2) the post intervention scores (posttest). The samples are related in the sense that each pretest has a corresponding posttest as both were obtained from the same student.
If the score of each student (ith) before and after the program is xi and yi respectively, then the pair (xi, yi) corresponds to the same subject (student in this case).
This is what is meant by paired sample. It is very important that two scores for each individual student be correctly identified and labeled as the differences di =│ xi - yi │are used to determine the test statistic and consequently the p-value.
However nothing concrete can be interpreted from it - specifically as to whether intervention program did have an impact as it could very well have happened by sheer luck (even though the students were drawn randomly) that for this sample of students, the scores did not change much.
On the other hand, it could also be the case that the program was indeed useful.
On the other hand if the null hypothesis is accepted, one can conclude that there is no evidence to suggest the program did have an impact.
This test has a few background assumptions which need to be satisfied.
It has however been shown that minor departures from normality do not affect this test - this is indeed an advantage.
This test is a small sample test. It is difficult to draw the 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 (below 30).