Two-sample proportion tests

Hypothesis Testing in R

Richie Cotton

Data Evangelist at DataCamp

Comparing two proportions

$H_{0}$: The proportion of SO users who are hobbyists is the same for those under thirty as those at least thirty.

$H_{0}$: $p_{\geq30} - p_{<30} = 0$

$H_{A}$: The proportion of SO users who are hobbyists is different for those under thirty as those at least thirty.

$H_{A}$: $p_{\geq30} - p_{<30} \neq 0$

alpha <- 0.05

Calculating the z-score

$$ z = \frac{(\hat{p}_{\geq30} - \hat{p}_{<30}) - 0}{\text{SE}(\hat{p}_{\geq30} - \hat{p}_{<30})} $$

$$ \text{SE}(\hat{p}_{\geq30} - \hat{p}_{<30}) = \sqrt{\dfrac{\hat{p} \times (1 - \hat{p})}{n_{\geq30}} + \dfrac{\hat{p} \times (1 - \hat{p})}{n_{<30}}} $$

$\hat{p}$ is a pooled estimate for $p$ (common unknown proportion of successes).

$$ \hat{p} = \frac{n_{\geq30} \times \hat{p}_{\geq30} + n_{<30} \times \hat{p}_{<30}}{n_{\geq30} + n_{<30} } $$

We only need to calculate 4 numbers: $\hat{p}_{\geq30}$, $\hat{p}_{<30}$, $n_{\geq30}$, $n_{<30}$.

Getting the numbers for the z-score

stack_overflow %>%
  group_by(age_cat) %>%
  summarize(
    p_hat = mean(hobbyist == "Yes"),
    n = n()
  )

# A tibble: 2 x 3
  age_cat     p_hat     n
  <chr>       <dbl> <int>
1 At least 30 0.773  1050
2 Under 30    0.843  1216

z_score

-4.217

Proportion tests using prop_test()

library(infer)
stack_overflow %>% 
  prop_test(

    hobbyist ~ age_cat,                   # proportions ~ categories

    order = c("At least 30", "Under 30"), # which p-hat to subtract

    success = "Yes",                      # which response value to count proportions of

    alternative = "two-sided",            # type of alternative hypothesis

    correct = FALSE                       # should Yates' continuity correction be applied?

  )

# A tibble: 1 x 6
  statistic chisq_df   p_value alternative lower_ci upper_ci
      <dbl>    <dbl>     <dbl> <chr>          <dbl>    <dbl>
1      17.8        1 0.0000248 two.sided     0.0605    0.165

Let's practice!

Hypothesis Testing in R