2.2 Carrying out the test: Two-sample test of proportions

We are now ready to carry out the one-sample test of proportions to test whether the proportion of social media users who use Facebook more than once per day is different from 0.73. The results o the test are as follows:

    2-sample test for equality of proportions with continuity correction

data:  c(154, 320) out of c(220, 416)
X-squared = 3.2776, df = 1, p-value = 0.07023
alternative hypothesis: two.sided
95 percent confidence interval:
 -0.145548179  0.007086641
sample estimates:
   prop 1    prop 2 
0.7000000 0.7692308 

We note the following:

  • The test statistic is equal to 3.2776
  • \(p\)-value is equal to 0.07023. Since this is larger than \(\alpha = 0.05\), we cannot reject \(H_0\). However, we can say that this is 'close to significant' since the \(p\)-value is not too much greater than 0.05 (i.e. it is less than 0.1)
  • The 95% confidence interval for the difference between \(p_1\) and \(p_2\) is (-0.1455, 0.0071), meaning that we are 95% confident that the difference in proportion between groups is within the interval (-0.1455, 0.0071). Since the interval includes 0, we cannot reject \(H_0\) at the \(\alpha = 0.05\) level of significance.
  • The sample proportions are \(\hat{p}_1 = 0.7\) and \(\hat{p}_2 = 0.7692\) (under prop 1 and prop 2).

We can, in fact, carry out hypothesis tests to test for differences in proportions between more than two groups. We will be considering such hypothesis tests in the next topic.