## 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.