145 Distribution of Differences in Sample Proportions (4 of 5)
145 Distribution of Differences in Sample Proportions (4 of 5)
Learning Objectives
- Draw conclusions about a difference in population proportions from a simulation.
The Sampling Distribution of Differences in Sample Proportions
Let’s summarize what we have observed about the sampling distribution of the differences in sample proportions. We want to create a mathematical model of the sampling distribution, so we need to understand when we can use a normal curve. We also need to understand how the center and spread of the sampling distribution relates to the population proportions.
Shape:
In each situation we have encountered so far, the distribution of differences between sample proportions appears somewhat normal, but that is not always true. We discuss conditions for use of a normal model later.
Center:
Regardless of shape, the mean of the distribution of sample differences is the difference between the population proportions, p1 – p2. This is always true if we look at the long-run behavior of the differences in sample proportions.
Spread:
We have observed that larger samples have less variability. Advanced theory gives us this formula for the standard error in the distribution of differences between sample proportions:
[latex]\sqrt{\frac{{p}_{1}(1-{p}_{1})}{{n}_{1}}+\frac{{p}_{2}(1-{p}_{2})}{{n}_{2}}}[/latex]
Notice the following:
- The terms under the square root are familiar. These terms are used to compute the standard errors for the individual sampling distributions of [latex]{\stackrel{ˆ}{p}}_{1}[/latex] and [latex]{\stackrel{ˆ}{p}}_{2}[/latex] .
- The sample size is in the denominator of each term. As we learned earlier this means that increases in sample size result in a smaller standard error.
Comment
Let’s look at the relationship between the sampling distribution of differences between sample proportions and the sampling distributions for the individual sample proportions we studied in Linking Probability to Statistical Inference. We compare these distributions in the following table.
Notice the relationship between the means:
- The mean of the differences is the difference of the means. This makes sense. The mean of each sampling distribution of individual proportions is the population proportion, so the mean of the sampling distribution of differences is the difference in population proportions.
Notice the relationship between standard errors:
- The standard error of differences relates to the standard errors of the sampling distributions for individual proportions. Look at the terms under the square roots. Since we add these terms, the standard error of differences is always larger than the standard error in the sampling distributions of individual proportions. In other words, there is more variability in the differences.
Variability and Variance
In this module, we sample from two populations of categorical data, and compute sample proportions from each.
We have seen that the means of the sampling distributions of sample proportio