Most real-world statistical problems involve trying to use sample statistics to make inferences about unknown parameters. This chapter is a prelude, in which we describe the distribution of a sample mean when the population mean and standard deviation are known parameters.

Suppose that:

1. The population mean and standard deviation are known, as m and s, respectively..
2. We take a random sample of n observations from the population.

What is the standard deviation of the sample mean, x? The main thing that we learn in this chapter is that the standard deviation of the sample mean, s_x, is equal to s divided by the square root of n.

For problems involving the sample mean, the Z transformation becomes

Using this equation, there are two types of questions that we can ask.

1. Given a specific sample size (say, 500), what is the probability that the sample mean will fall within a certain range? For example, what is the probability that the sample mean will be less than 20? What is the probability that the sample mean will be between 0.5 and 3.5? What is the probability that the sample mean will be greater than 0?

   To answer this question, we calculate a value for Z and use normcdf(). This is like the problems from chapter 2.

2. What is the sample size, n, needed to result in a probability no greater than y that the sample mean will fall within a certain range? That is, instead of being given n and asked to compute a probability, we are given a probability and asked for n.

   In this case, we use invnorm(y) to find the value of Z. Then, we have Z, X, m, and s, so all that we need to solve for is n.

Finally, there are two ways to arrive at the population standard deviation, s. In anything other than a binomial setting, it has to be given to us. In a binomial setting, we can compute it as the square root of p(1-p).