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 proportions of smokers
\(H_a\): the 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
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 is questionable if the differences in the proportions have any practical meaning.