Lesson 2

Let us explain this quantitatively. Suppose we want to achieve the following guarantee: \[\Pr\left[\left|\frac{\alpha}{n}-p\right| \le 0.04\right] \ge 0.95 \ .\] That is, with probability at least \(0.95\), our estimate deviates from the actual value by no more than \(0.04\). The segment \([\alpha/n-0.04, \alpha/n+0.04]\) is called the confidence interval. (In parentheses, we note that we could not require inclusion in the confidence interval with unit probability: for a reasonably sized sample, there is always a nonzero probability that we were very unlucky with the sample.)