Estimating Population Proportions

We use the prop.test command to construct confidence interval estimates for population proportions, and to test claims about proportions.  The difference will be the arguments we include.

To construct a confidence interval estimate for population proportions we need to identify values for the following arguments:

• x, the number of successes
• n, the sample size
• conf.level, the confidence level for our estimate (expressed as a decimal)

1-Sample

Example: In the 2012 Summer Olympic Games 85 countries competed.  Of the countries who competed, 50 won at least one gold medal.  Construct a 95% confidence interval estimate for the population proportion of countries who win at least one gold medal in the Summer Olympics.
Source: wikipedia.org

> prop.test(x = 50, n = 85, conf.level = .95)

        1-sample proportions test with continuity correction

data:  50 out of 85, null probability 0.5 
X-squared = 2.3059, df = 1, p-value = 0.1289
alternative hypothesis: true p is not equal to 0.5 
95 percent confidence interval:
 0.4761797 0.6922411 
sample estimates:
        p 
0.5882353 

A fair amount of information is being output, but in this instance we are only concerned with the lines that tell us the 95 percent confidence interval estimate is 0.476797 to 0.6922411.

2-Sample

For a 2-sample confidence interval we will use the same arguments as above, but we will have two values for both x and n.  To properly enter these arguments we write x = c(x₁, x₂) and n = c(n₁, n₂) so that x and n are both lists composed of two values.

Example: In 2005 there were 28,534 total infant deaths in America (ages 0-1).  Of these, 6.3% were due to maternal complications, and 4% were due to accidents or unintentional injuries.  Construct a 93% confidence interval estimate for the difference in the population proportions.
Source: www.nhpco.org

In this case we are not given values for x₁ and x₂ so we will have to calculate them as x₁ = n₁ · p̂₁ and x₂ = n₂ · p̂₂, with both values rounded to the nearest whole number.  We can use the round command to complete these calculations in one step.

> x.one = round(28534 * 0.063)
> x.one
[1] 1798
> x.two = round(28534 * 0.04)
> x.two
[1] 1141

Now that we have x₁ = 1798 and x₂ = 1141 we can construct the confidence interval.

> prop.test(x = c(1798, 1141), n = c(28534, 28534), conf.level = .93)

        2-sample test for equality of proportions with continuity correction

data:  c(1798, 1141) out of c(28534, 28534) 
X-squared = 154.3728, df = 1, p-value < 2.2e-16
alternative hypothesis: two.sided 
93 percent confidence interval:
 0.01964198 0.02640834 
sample estimates:
    prop 1     prop 2 
0.06301255 0.03998738 

So the 93 percent confidence interval estimate for p₁ - p₂ is 0.01964198 to 0.02640834.  Thus, the data suggests that p₁ > p₂.