146 Distribution of Differences in Sample Proportions (5 of 5)
146 Distribution of Differences in Sample Proportions (5 of 5)
Learning Objectives
- Estimate the probability of an event using a normal model of the sampling distribution.
Why Do We Care about a Normal Model?
Now we focus on the conditions for use of a normal model for the sampling distribution of differences in sample proportions.
We use a normal model for inference because we want to make probability statements without running a simulation. If we are conducting a hypothesis test, we need a P-value. If we are estimating a parameter with a confidence interval, we want to state a level of confidence. These procedures require that conditions for normality are met.
Note: If the normal model is not a good fit for the sampling distribution, we can still reason from the standard error to identify unusual values. We did this previously. For example, we said that it is unusual to see a difference of more than 4 cases of serious health problems in 100,000 if a vaccine does not affect how frequently these health problems occur. But without a normal model, we can’t say how unusual it is or state the probability of this difference occurring.
When Is a Normal Model a Good Fit for the Sampling Distribution of Differences in Proportions?
A normal model is a good fit for the sampling distribution of differences if a normal model is a good fit for both of the individual sampling distributions. More specifically, we use a normal model for the sampling distribution of differences in proportions if the following conditions are met.
[latex]{n}_{1}{p}_{1}≥10\text{ }{n}_{1}(1-{p}_{1})≥10\text{ }{n}_{2}{p}_{2}≥10\text{ }{n}_{2}(1-{p}_{2})≥10[/latex]
These conditions translate into the following statement:
The number of expected successes and failures in both samples must be at least 10. (Recall here that success doesn’t mean good and failure doesn’t mean bad. A success is just what we are counting.)
Here we complete the table to compare the individual sampling distributions for sample proportions to the sampling distribution of differences in sample proportions.
Example
More on Conditions for Use of a Normal Model
All of the conditions must be met before we use a normal model. If one or more conditions is not met, do not use a normal model. Here we illustrate how the shape of the individual sampling distributions is inherited by the sampling distribution of differences.
Learn By Doing
Recall the AFL-CIO press release from a previous activity. “Fewer than half of Wal-Mart workers are insured under the company plan – just 46 percent. This rate is dramatically lower than the 66 percent of workers at large private firms who are insured under their companies’ plans, according to a new Commonwealth Fund study released today, which documents the growing trend among large employers to drop health insurance for their workers.”
Learn By Doing
Using the Normal Model in Inference
When conditions allow the use of a normal model, we use the normal distribution to determine P-values