Statistical Inference: A Short Course

9.1 The Sampling Distribution of a Proportion

Our goal in this section is to estimate the parameter p of a binomial population. Remember that a binomial population is one whose elements belong to either of two classes—success or failure. Previously, the binomial random variable X was defined as the number of successes obtained in n independent trials of a simple alternative experiment and p denoted the probability of a success. Now, X will be taken to be the number of successes observed in a simple random sample of size n and p will represent the proportion of successes in the population. (Obviously, 1 − p is the population proportion of failures.)

The problem that we now face is the estimation of p. Also, since we are basing our estimate of p on sample information, we need to obtain some idea of the error involved in estimating p. In order to adequately address these issues, we need to examine the characteristics of the sampling distribution of the sample proportion. To this end, let us extract a random sample of size n from a binomial population and find the observed relative frequency of a success

(9.1)

As we shall see shortly, serves as our “best” estimate for p. Now, we know that is a random variable that varies under random sampling, depending upon which sample is chosen. ...