Chapter 5.5, Problem 65E

Suppose that when the pH of a certain chemical compound is 5.00, the pH measured by a randomly selected beginning chemistry student is a random variable with mean 5.00 and standard deviation .2. A large batch of the compound is subdivided and a sample given to each student in a morning lab and each student in an afternoon lab. Let $\bar{X}$ = the average pH as determined by the morning students and $\bar{Y}$= the average pH as determined by the afternoon students.

a. If pH is a normal variable and there are 25 students in each lab, compute $P(-1\le \bar{X}-\bar{Y}\le 1)$ [Hint: $\bar{X}-\bar{Y}$ is a linear combination of normal variables, so is normally distributed. Compute ${\mu }_{\bar{X}-\bar{Y}}\text{and}{\sigma }_{\bar{X}-\bar{Y}}$_.]

b. If there are 36 students in each lab, but pH determinations are not assumed normal, calculate (approximately) $P(-.1\le \bar{X}-\bar{Y}\le .1).$