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*,

(12.3) |

which is simply the average of over the subjects, and the

The variance estimate (12.4) is considered to be a

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 [125] and other books on mathematical statistics for these and many more.

The *Student's t test* [319] involves calculating the following:

(12.5) |

in which

(12.6) |

and is the number of subjects who received treatment . The subtractions by and in the expressions are due to Bessel's correction. Based on the value of , the confidence in the null hypothesis is determined by looking in a table of the Student's t cdf (Figure 12.5(b)). Typically, or lower is sufficient to declare that is true (corresponding to 95% confidence). Such tables are usually arranged so that for a given and is, the minimum value needed to confirm with confidence is presented. Note that if is negative, then the effect that has on runs in the opposite direction, and is applied to the table.

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 [351]. 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 [125].

Steven M LaValle 2020-11-11