There are two possible cases when testing two population means, the dependent case and the
independent case. Most books treat the independent case first, but I'm putting the dependent
case first because it follows immediately from the test for a single population mean in the
The Mean of the Difference:
The idea with the dependent case is to create a new variable, D, which is the difference between
the paired values. You will then be testing the mean of this new variable.
Here are some steps to help you accomplish the hypothesis testing
- Write down the original claim in simple terms. For example: After > Before.
- Move everything to one side: After - Before > 0.
- Call the difference you have on the left side D: D = After - Before > 0.
- Convert to proper notation:
- Compute the new variable D and be sure to follow the order you have defined in step 3. Do
not simply take the smaller away from the larger. From this point, you can think of having a
new set of values. Technically, they are called D, but you can think of them as x. The
original values from the two samples can be discarded.
- Find the mean and standard deviation of the variable D. Use these as the values in the t-test
from the single mean test.
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