142 Distribution of Differences in Sample Proportions (1 of 5)
142 Distribution of Differences in Sample Proportions (1 of 5)
Learning Objectives
- Describe the sampling distribution of the difference between two proportions.
- Draw conclusions about a difference in population proportions from a simulation.
Our goal in this module is to use proportions to compare categorical data from two populations or two treatments.
It’s not about the values – it’s about how they are related!
In Inference for One Proportion, we learned to estimate and test hypotheses regarding the value of a single population proportion. Here, in Inference for Two Proportions, the value of the population proportions is not the focus of inference. Instead, we want to develop tools comparing two unknown population proportions.
The first step is to examine how random samples from the populations compare. In this investigation, we assume we know the population proportions in order to develop a model for the sampling distribution. This is the same thinking we did in Linking Probability to Statistical Inference. In that module, we assumed we knew a population proportion. Then we selected random samples from that population. We examined how sample proportions behaved in long-run random sampling. This is the same approach we take here.
Example
Teen Depression
Most of us get depressed from time to time. Depression is a normal part of life. Many people get over those feelings rather quickly. But some people carry the burden for weeks, months, or even years. For these people, feelings of depression can have a major impact on their lives. Depression can cause someone to perform poorly in school or work and can destroy relationships between relatives and friends.
Research suggests that teenagers in the United States are particularly vulnerable to depression. And, among teenagers, there appear to be differences between females and males. The Christchurch Health and Development Study (Fergusson, D. M., and L. J. Horwood, “The Christchurch Health and Development Study: Review of Findings on Child and Adolescent Mental Health,” Australian and New Zealand Journal of Psychiatry 35[3]:287–296), which began in 1977, suggests that the proportion of depressed females between ages 13 and 18 years is as high as 26%, compared to only 10% for males in the same age group.
Let’s assume that 26% of all female teens and 10% of all male teens in the United States are clinically depressed. In other words, assume that these values are both population proportions.
- pf = 0.26 for the population of all female teenagers in the United States
- pm = 0.1 for the population of all male teenagers in the United States
Graphically, we can compare these proportion using side-by-side ribbon charts:
To compare these proportions, we could describe how many times larger one proportion is than the other. Here the female proportion is 2.6 times the size of the male proportion (0.26/0.10 = 2.6). An easier way to compare the proportions is to simply subtract them. This is the approach stati