148 Estimate the Difference between Population Proportions (1 of 3)
148 Estimate the Difference between Population Proportions (1 of 3)
Learning Objectives
- Recognize when to use a hypothesis test or a confidence interval to compare two population proportions or to investigate a treatment effect for a categorical variable.
- Construct a confidence interval to estimate the difference between two population proportions (or the size of a treatment effect) when conditions are met. Interpret the confidence interval in context.
In “Distributions of Differences in Sample Proportions,” we used simulation to observe the behavior of the differences in sample proportions when we randomly select many, many samples. From the simulation, we developed a normal probability model to describe the sampling distribution of sample differences. With this model, we are now ready to do inference about a difference in population proportions (or about a treatment effect.)
When our goal is to estimate a difference between two population proportions (or the size of a treatment effect), we select two independent random samples and use the difference in sample proportions as an estimate. Of course, random samples vary, so we want to include a statement about the amount of error that may be present. Because the differences in sample proportions vary in a predictable way, we can also make a probability statement about how confident we are in the process that we used to estimate the difference between the population proportions. You may recognize that what we are describing is a confidence interval.
In Inference for One Proportion, we calculated confidence intervals to estimate a single population proportion. In this section, “Estimate the Difference between Population Proportions,” we learn to calculate a confidence interval to estimate the difference between two population proportions. If the data comes from an experiment, we estimate the size of the treatment effect.
Learn By Doing
Confidence Interval for a Difference in Two Population Proportions: the Basics
Every confidence interval has this form:
[latex]\mathrm{statistic}\text{}±\text{}\mathrm{margin}\text{}\mathrm{of}\text{}\mathrm{error}[/latex]
To estimate a difference in population proportions (or a treatment effect), the statistic is a difference in sample proportions, so the confidence interval is
[latex](\mathrm{difference}\text{}\mathrm{in}\text{}\mathrm{sample}\text{}\mathrm{proportions})\text{}±\text{}\mathrm{margin}\text{}\mathrm{of}\text{}\mathrm{error}[/latex]
When we select two random samples and calculate the difference in the sample proportions, we do not know the exact amount of error for this particular pair of samples. We therefore use the standard error as a typical amount of error and calculate the margin of error from the standard error, as we did in Inference for One Proportion.
If a normal model is a good fit for the sampling distribution, we can use it to make probability statements that describe our confidence in the interval. More specifically, we use the normal