1. QBM117 - Business Statistics Estimating the population proportion

2. Objectives w To give some examples where we need to estimate the population proportion. w To describe the sampling distribution of the sample proportion, w To derive the confidence interval estimate of the population proportion.

3. Estimating the population proportion When the random variable of interest is qualitative, the characteristic we usually consider is the proportion of successes. For example, we may be interested in the proportion of homes for sale, that have a modern kitchen. The proportion of voters who will vote for a particular political candidate

4. In the topic on probability distributions, we saw that the number of successes in n trials followed a binomial distribution. Now, instead of expressing the number of successes X, we can easily convert this variable to the proportion of successes, by dividing by the sample size, n. Therefore, the point estimator of the population proportion p, is the sample proportion, p hat,

5. The sampling distribution of The sample proportion is approximately normal with mean p and standard deviation provided that Therefore the standardised random variable is approximately standard normally distributed

6. Finding an interval estimate of the population proportion, p The interval estimate of p is found by however, since we are trying to estimate p, we cannot determine therefore we must estimate p and q by Therefore the confidence interval estimator of p is found by as long as

7. Exercise 1 – Exercise 8.27 p270 w Here we want to estimate the population proportion, p. w The sample proportion is the best estimator of p. will be approximately normally distributed. Now since Therefore the confidence interval estimator of p is found by

8. Therefore the confidence interval is given by

9. Example 2 w In a random sample of 50 homeowners selected from a large suburban area, 19 said that they had serious problems with excessive noise from their neighbours. w Find a 99% confidence interval estimate of the percentage of all homeowners in this suburban area who have such problems.

10. Reading for next lecture w S&S Chapter 8 Sections 8.6 - 8.7 Exercises to be completed before next lecture w S&S 8.35