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 proportion of US adults aged 18-29 who say they use Facebook
  • \(p_2\) denotes the 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 secong sample \(x_2\) individuals exhibit the trait of interest. The estimated 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.

References

Auxier, Brooke, and Monica Anderson. 2021. “Social Media Use in 2021.” Pew Research Center. 2021. https://www.pewresearch.org/internet/2021/04/07/social-media-use-in-2021/.