Student’s t-tests
Recall that when we know µ and σ for the population, we can determine the probability that our sample came from that population with a z-test.
What if you do not know µ and σ?
One-sample t-test: know µ but not σ.
Looks just like the formula for z-test, but you are substituting an estimate of σ (s). This estimate almost certainly underestimates the actual σ because it is based on variability around the sample mean M and not the actual population mean µ. The distribution of all possible t-scores (that you use to assess the probability) will depend on how accurate the M ( is as an estimate of σ, which is dependent on the size of the sample n. As n approaches ∞ the t distribution will approach z. For other ns you need to consult the propoer t distribution, based on degrees of freedom. For this one-sample t-test, df
= n-1.
Two-sample independent t-test: know nothing about the underlying population.
You use a control group (here Group 2) to offer an estimate of both µ and σ of the underlying population if you have done nothing to the participants, and then ask if the experimental group
(Group 1) differs. The degrees of freedom, df, = n
1
+ n
2
-2.

Two-sample matched t-test: comparing differences in the same (or paired) people).
If there is some basis for matching the scores
(same participant tested in two conditions, or different participants matched across conditions), then you calcualte a difference score for each pair, and calculate a t just like the one-sample t test, where you are asking if the mean difference score differs from the known mean difference when there is no effect. Degrees of freedom, df, = n
D
– 1.