There are two possible cases whenever testing two population means: dependent case and the independent case. Most of the books at first treat the independent, but we are going to discuss the dependent case first as it follows instantly from the test for a single population mean.
Mean of the Difference: μD
The basic idea with dependent case is to make a new variable D that is the difference between the paired values. We will then test the mean of new variable.Some of the steps are shown below to accomplish the hypothesis testing:
A) A first write down the original claim in simple terms. For illustration: After > Before.
B) Move the whole thing to one side: After - Before > 0.
C) Call the difference you encompass on left side D: D = After - Before > 0.
D) Transform to proper notation: μD > 0
E) Calculate the new variable D and be sure to pursue the order you have stated in step 3. Do not simply take smaller away from the bigger. From this point, you can think of containing a new set of values. Technically, they are termed D, however you can think of them as x. The original values from two samples can be rejected.
F) Determine the mean and standard deviation of the variable D. We can use these as the values in t-test.
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