Inference for Proportions

Proportions Our earlier analysis focused on inference about population means. Now we turn our attention to inference about the proportion of some outcome in a population. We will consider a single population and then compare proportions from two populations or treatments.

The Basics Population Proportion: Sample Proportion: is a point estimate of P

Sampling Distribution of Sample Proportion Choose an SRS of size n from a population that contains proportion p of ‘characteristic’. Let the sample proportion be defined as the number in the sample with the ‘characteristic’ divided by n. As the sample size increases, the sampling distribution of the sample proportion becomes approximately normal. The mean of the sampling distribution is p, the population proportion. The standard deviation of the sampling distribution is

Assumptions for Inference about a Proportion The data are an SRS from the population of interest. The population is at least ’10’ times as large as the sample. Fact: The normal approximation to the distribution of the sample proportion is most accurate when p = 0.5

Let’s Try Some Examples

Comparing Two Proportions Setting 1: Independent Samples from two populations proportion of items in population 1 with characteristic proportion of items in population 2 with characteristic   Setting 2: Randomly assign subjects to one of two treatments Probability of success with treatment 1 Probability of success with treatment 2 In both settings, we wish to compare .

The Data

For large

Hypothesis Test for Ho: Under Ho, estimate the same quantity.

Test Statistic

Set Up for Ha

How about another example?