Question

What are the similarities and differences between the two t tests for the difference between two population means? Under what condition should we use the t test for the matched samples?

Answer #1

**Solution :**

The two-sample *t*-test is used when the data of two
samples are statistically independent, while the paired
*t*-test is used when data is in the form of matched
pairs.

To use the two-sample *t*-test, we need to assume that
the data from both samples are normally distributed and they have
the same variances. For paired *t*-test, we only require
that the difference of each pair is normally distributed. An
important parameter in the *t*-distribution is the degrees
of freedom.

For two independent samples with equal sample size n,
*df* = 2(*n*-1) for the two-sample *t*-test.
However, if we have *n* matched pairs, the actual sample
size is *n* (pairs) although we may have data from
2*n* different subjects. The paired *t*-test is, in
fact, a one-sample *t*-test, which makes its *df* =
n-1.

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