Module 7: Linking Probability to Statistical Inference
Module 7: Linking Probability to Statistical Inference
Distribution of Sample Proportions (5 of 6)
Distribution of Sample Proportions (5 of 6)
Learning OUTCOMES
- Use a z-score and the standard normal model to estimate probabilities of specified events.
From our work on the previous page, we now have a mathematical model of the sampling distribution of sample proportions. This model describes how much variability we can expect in random samples from a population with a given parameter. If a normal model is a good fit for a sampling distribution, we can apply the empirical rule and use z-scores to determine probabilities. Here we link probability to the kind of thinking we do in inference.
Making Connections to Probability Models in Probability and Probability Distribution
Probability describes the chance that a random event occurs. Recall the concept of a random variable from the module Probability and Probability Distribution. When a variable is random, it varies unpredictably in the short run but has a predictable pattern in the long run. Sample proportions from random samples are a random variable. We cannot predict the proportion for any one random sample; they vary. But we can predict the pattern that occurs when we select a great many random samples from a population. The sampling distribution describes this pattern. When a normal model is a good fit for the sampling distribution, we can use what we learned in the previous module to find probabilities.
Recall probability models we saw in Probability and Probability Distribution. We saw examples of models with skewed curves, but we focused on normal curves because we use normal probability models to describe sampling distributions in Modules 7 to 10 when we make inferences about a population. As we now know, we can use a normal model only when certain conditions are met. Whenever we want to use a normal model, we must check the conditions to make sure a normal model is a good fit.
Here we summarize our general process for developing a probability model for inference. This is essentially the same process we used in the previous module for developing normal probability models from relative frequencies.
If a normal model is a good fit for the sampling distribution, we can standardize the values by calculating a z-score. Then we can use the standard normal model to find probabilities, as we did in Probability and Probability Distribution.
The z-score is the error in the statistic divided by the standard error. For sample proportions, we have the following formulas.
[latex]\text{standard error} = \sqrt{\frac{p(1 - p)}{n}}[/latex]
[latex]Z = \frac{\text{statistic} - \text{parameter}}{\text{standard error}} = \frac{\hat{p} - p}{\text{standard error}}[/latex]
We can also write this as one formula:
[latex]Z = \frac{\hat{p} - p}{\sqrt{\frac{p(1 - p)}{n}}}[/latex]
Comment
This z-score formula is similar to the z-score formula we used in Probability and Probability Distribution. We described the z-score