Hypothesis Testing for Two Population Proportions
In this section, we will step through how to perform a hypothesis tests to determine if there is a difference between two population proportions.
Distribution Used
We will continued to use the Normal Distributions and z-scores.
![Image with multiple bell shaped curves. Two of the curves have a population mean at zero and the standard deviation varies. The curve with the smaller standard deviation is less spread out and more concentrated around the middle. The curve with the mean at one is displaced to the right by one unit. It also has a larger standard deviation and is, therefore, more spread out.](https://pressbooks.bccampus.ca/1130sandbox/wp-content/uploads/sites/2128/2024/02/NormalDistributions.jpg)
Assumptions
In order to be able to perform the analysis in this section, the following assumptions must hold true:
- the samples are random and independent of one another
- the sample sizes and proportions from each group are large enough such that:
- [latex]np > 5[/latex]
- [latex]n(1-p) > 5[/latex]
Note: When the sample size and proportion are large enough, the discrete distribution approaches a normal/bell shaped curve:
![Binomial Distribution Graphs for Different Values of p](https://pressbooks.bccampus.ca/1130sandbox/wp-content/uploads/sites/2128/2023/12/Binomial_Graphs.jpg)
The Difference Between the Two Curves Above
For the left-most curve:
- [latex]np = 50\times 0.5 = 25 > 5[/latex]
- [latex]n(1-p) = 50 \times (1-0.5) = 25 > 5[/latex]
- the curve closely resembles a bell-shaped curve
For the right-most curve:
- [latex]np = 50\times 0.9 = 45 > 5[/latex]
- [latex]n(1-p) = 50 \times (1-0.9) = 5 \ngtr 5[/latex]
- the curve is skewed left and therefore not bell-shaped
Note: The right-most curve is at the ‘limit’ of acceptable. If the value of [latex]n[/latex] was slightly larger or the value of [latex]p[/latex] slightly smaller (to increase the size of [latex]1-p[/latex]), we could perform the analysis in this section on this data.