## What does this test do?

• decides if two (or more) proportions differ significantly
• quite similar task as for fisher.test() but allows for larger numbers due to some approximations made
• $$H_0$$ : $$p_1 = p_2 = \dots = p_k$$ (equal proportions, independence)
• a small p-value means that the null is rejected, i.e. that the proportions differ significantly
• function: prop.test()

## Running the test

smokers  <- c( 83, 90, 129, 70 )
patients <- c( 86, 93, 136, 82 )
percent_smokers = round(smokers/patients*100, 1)
percent_smokers 
## [1] 96.5 96.8 94.9 85.4
barplot(percent_smokers, col = "red")

$$H_0$$: the 4 populations from which the patients were drawn have the same true proportion of smokers
$$H_a$$: this proportion is different in at least one of the populations

prop.test(smokers, patients) 
##
##  4-sample test for equality of proportions without continuity
##  correction
##
## data:  smokers out of patients
## X-squared = 12.6, df = 3, p-value = 0.005585
## alternative hypothesis: two.sided
## sample estimates:
##    prop 1    prop 2    prop 3    prop 4
## 0.9651163 0.9677419 0.9485294 0.8536585

## Reading the output

The null hypothesis is rejected, even if one would not expect this looking at the barplot. Significance is achieved because of the large sample sizes. This is a general issue when it comes to hypothesis tests. It it questionable if the differences in the proportions have any practical meaning.