Chapter 2 Two-sample test of proportions
If we wish to compare two proportions from different (independent) populations, we can use the two-sample test of proportions. For example, suppose we wish to know whether there is a significant difference in the proportion of US adults who say they use Facebook between two groups: those aged 18-29, and those aged 30-49. Consider the hypotheses
\[H_0 : p_1 = p_2 \text{ versus } H_1 : p_1 \neq p_2,\]
where:
- \(p_1\) denotes the population proportion of US adults aged 18-29 who say they use Facebook
- \(p_2\) denotes the population proportion of US adults aged 30-49 who say they use Facebook
For a two-sample test of proportions, we also have that the sample sizes chosen from each population (or group) are \(n_1\) and \(n_2\) respectively, and that in the first sample, \(x_1\) individuals exhibit the trait of interest, and in the second sample \(x_2\) individuals exhibit the trait of interest. The estimated (or sample) proportions are
\[\hat{p}_1 = \frac{x_1}{n_1} \text{ and } \hat{p}_2 = \frac{x_2}{n_2}.\]
A survey was carried out (Auxier and Anderson 2021) to better understand Americans' use of social media, online platforms, and messaging apps. Supposing that of the \(n_1 = 220\) and \(n_2 = 416\) respondents from each group respectively, \(x_1 = 154\) and \(x_2 = 320\) said they used Facebook, we have that
- \(\hat{p}_1 = \displaystyle \frac{x_1}{n_1} = \frac{154}{220} = 0.7\)
- \(\hat{p}_2 = \displaystyle \frac{x_2}{n_2} = \frac{320}{416} = 0.77\).
From these sample proportions, we can see that 70% of 18-29 year olds say they use Facebook, compared with 77% of 30-49 year olds. In the following sections, we will see whether this difference is statistically significant.