Unfortunately, the scientist is not able to perform the same experiment at the same time on all people. She must instead draw a small set of people from the population and make a determination about whether the hypothesis is true. Let the index refer to a particular chosen subject, and let be his or her response for the experiment; each subject's response is a dependent variable. Two statistics are important for combining information from the dependent variables: The mean,
To test the hypothesis, Student's t-distribution (``Student'' was William Sealy Gosset) is widely used, which is a probability distribution that captures how the mean is distributed if subjects are chosen at random and their responses are averaged; see Figure 12.5. This assumes that the response for each individual is a normal distribution (called Gaussian distribution in engineering), which is the most basic and common probability distribution. It is fully characterized in terms of its mean and standard deviation . The exact expressions for these distributions are not given here, but are widely available; see  and other books on mathematical statistics for these and many more.
The Student's t test  involves calculating the following:
The binary outcome might not be satisfying enough. This is not a problem because difference in means, , is an estimate of the amount of change that applying had in comparison to . This is called the average treatment effect. Thus, in addition to determining whether the is true via the t-test, we also obtain an estimate of how much it affects the outcome.
Student's t-test assumed that the variance within each group is identical. If it is not, then Welch's t-test is used . Note that the variances were not given in advance in either case. They are estimated ``on the fly'' from the experimental data. Welch's t-test gives the same result as Student's t-test if the variances happen to be the same; therefore, when in doubt, it may be best to apply Welch's t-test. Many other tests can be used and are debated in particular contexts by scientists; see .
Steven M LaValle 2020-01-06