Suppose in that we wished to design a test so that if the pH were really equal to 8.20, then this conclusion will be reached with probability equal to .95. On the other hand, if the pH differs from 8.20 by .03 (in either direction), we want the probability of picking up such a difference to exceed .95. (a) What test procedure should be used? (b) What is the required sample size? (c) If x = 8.31, what is your conclusion? (d) If the actual pH is 8.32, what is the probability of concluding that the pH is not 8.20, using the foregoing procedure?

Lecture 13 Nicole Rubenstein October 17, 2017 De▯nition 1.1. The gamma function, denoted by ▯, is an improper integral de▯ned from 0 to 1, such that Z 1 ▯(z) = x z▯1e▯xdx: 0 Theorem 1.1. For z > 1, we have ▯(z) = (z ▯ 1)▯(z ▯ 1): To prove this theorem, we need to use integration by parts: De▯nition 1.2. Let u(x) and v(x) be functions of x. We can write integration by parts as follows: Z Z udv = uv ▯ vdu: R Example 1.1. Inte