Performing a test of equal proportions

For this R tutorial we will use the example from Penn State’s Introduction to Mathematical Statistics course. Assume that 800 American adults were polled on the question: “Should the federal tax on cigarettes be raised to pay for health care reform?”. Of those polled, 195 identified themselves as smokers, 41 of which responded ‘yes’, while 605 identified as non-smokers with 351 responding yes. Plotting the results of the survey (below) we might suggest that there is a difference in how smokers and non-smokers responded overall, so we can use a proportion test to statistically support or reject this hypothesis.

In this tutorial we will manually input the data into R. First, we will make two vectors, one with the total number of non-smokers and smokers polled (count) and the second with the number of yes responds for each group (yes) with the c() function.

n <- c(605, 195)
yes <- c(351, 41)

We can perform a proportion test using the prop.test() function which takes the vector of successes as the first argument and the total number of trials as the second argument.

prop.test(yes, n)

## 
##  2-sample test for equality of proportions with continuity correction
## 
## data:  yes out of n
## X-squared = 79.273, df = 1, p-value < 2.2e-16
## alternative hypothesis: two.sided
## 95 percent confidence interval:
##  0.2971087 0.4427091
## sample estimates:
##    prop 1    prop 2 
## 0.5801653 0.2102564

The p-value for our test is extremely small and well below 0.05, so we can conclude that there is a statistically significant difference between the two proportions. Specifically, the proportion of smokers who answered ‘yes’ to the survey question is significantly lower than the proportion of non-smokers who answered ‘yes’.

Note that the test statistic for our test is slightly different than in the STAT 415 example. This is because prop.test() uses the Yates’ continuity correction by default. By setting correct = FALSE within prop.test() the correction will not be made to result in the same test statistic that is calculated in the STAT 415 example.

Full code block

# Input the total number of non-smokers and somkers polled and the number of yes responses for each
n <- c(605, 195)
yes <- c(351, 41)

# Perform a proportion test
prop.test(yes, n)