The dependent t-test is testing the null hypothesis that there are no differences between the means of the two related groups. If we get a statistically significant result, we can reject the null hypothesis that there are no differences between the means in the population and accept the alternative hypothesis that there are differences between the means in the population. We can express this as follows:
Before we answer this question, we need to point out that you cannot choose one test over the other unless your study design allows it. What we are discussing here is whether it is advantageous to design a study that uses one set of participants whom are measured twice or two separate groups of participants measured once each. The major advantage of choosing a repeated-measures design (and therefore, running a dependent t-test) is that you get to eliminate the individual differences that occur between participants – the concept that no two people are the same – and this increases the power of the test. What this means is that if you are more likely to detect a (statistically significant) difference, if one does exist, using the dependent t-test versus the independent t-test.
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Yes, but this does not happen very often. You can use the dependent t-test instead of using the usual independent t-test when each participant in one of the independent groups is closely related to another participant in the other group on many individual characteristics. This approach is called a "matched-pairs" design. The reason we might want to do this is that the major advantage of running a within-subject (repeated-measures) design is that you get to eliminate between-groups variation from the equation (each individual is unique and will react slightly differently than someone else), thereby increasing the power of the test. Hence, the reason why we use the same participants – we expect them to react in the same way as they are, after all, the same person. The most obvious case of when a "matched-pairs" design might be implemented is when using identical twins. Effectively, you are choosing parameters to match your participants on, which you believe will result in each pair of participants reacting in a similar way.
You need to report the test as follows:
where df is N – 1, where N = number of participants.
