Suppose scientists invent a new drug that supposedly will inhibit a mouse’s ability to run through a maze. The scientists design an experiment in which three mice are randomly chosen to receive the drug and another three mice serve as controls by ingesting a placebo. The time each mouse takes to go through a maze is measured in seconds. Suppose the results of the experiment are as follows:
The average time for the drug group is 25 s and the average time for the control group is 20.33 s. The mean difference in times is 25 − 20.33 = 4.67 s.
The average time for the mice given the drug is greater than the average time for the control group, but this could be due to random variability rather than a real drug effect. We cannot, however, tell for sure whether there is a real effect. What we do instead is that we estimate how easily pure random chance would produce a difference this large. If that probability is small, then we conclude there is something other than pure random chance at work, and hence there is a real effect.
If the drug does not really influence times, then split of the six observations into two groups was essentially random. The outcomes could just as easily be distributed:
In this case, the mean difference ...