Difference Between Proportions
Statistics problems often involve comparisons between two
independent sample proportions. This lesson explains how to compute
probabilities associated with differences between proportions.
Difference Between Proportions: Theory
Suppose we have two
populations
with proportions equal to P_{1} and P_{2}. Suppose
further that we take all possible
samples
of size n_{1} and n_{2}. And finally, suppose that the
following assumptions are valid.
- The samples are
independent;
that is, observations in population 1 are not affected by observations
in population 2, and vice versa.
Given these assumptions, we know the following.
It is straightforward to derive the last bullet point, based on material
covered in previous lessons. The derivation starts with a recognition
that the variance of the difference between independent random variables is
equal to the sum of the individual variances. Thus,
σ^{2}_{d} =
σ^{2}_{P1} _{-} _{P2} =
σ^{2}_{1} + σ^{2}_{2}
If the populations N_{1} and N_{2} are both large
relative to n_{1} and n_{2}, respectively,
then
σ^{2}_{1} =
P_{1}(1 - P_{1}) / n_{1}
And
σ^{2}_{2} =
P_{2}(1 - P_{2}) / n_{2}
Therefore,
σ^{2}_{d} =
[ P_{1}(1 - P_{1}) / n_{1} ] +
[ P_{2}(1 - P_{2}) / n_{2} ]
And
σ_{d} =
sqrt{ [ P_{1}(1 - P_{1}) / n_{1} ] +
[ P_{2}(1 - P_{2}) / n_{2} ] }
Difference Between Proportions: Sample Problem
In this section, we work through a sample problem to show how to apply
the theory presented above. In this example,
we will use Stat Trek's
Normal Distribution Calculator
to compute probabilities.
Normal Distribution Calculator
The normal calculator solves common statistical problems, based on the normal
distribution. The calculator computes cumulative probabilities, based on three
simple inputs. Simple instructions guide you to an accurate solution, quickly
and easily. If anything is unclear, frequently-asked questions and sample
problems provide straightforward explanations. The
calculator is free. It can found in the Stat Trek
main menu under the Stat Tools tab. Or you can tap the button below.
Normal Distribution Calculator
Sample Problem
In one state, 52% of the voters are Republicans, and 48% are Democrats.
In a second state, 47% of the voters are Republicans, and 53% are
Democrats. Suppose 100 voters are surveyed from each state.
Assume the survey uses simple random sampling.
What is the probability that the survey
will show a greater percentage of Republican voters in the
second state than in the first state?
(A) 0.04
(B) 0.05
(C) 0.24
(D) 0.71
(E) 0.76
Solution
The correct answer is C. For this analysis, let P_{1} =
the proportion of Republican voters in the first state,
P_{2} = the proportion of Republican voters in the second state,
p_{1} = the proportion of Republican voters in the
sample from the first state, and
p_{2} = the proportion of Republican voters in the
sample from the second state. The number of voters sampled from
the first state (n_{1}) = 100, and the number of voters
sampled from the second state (n_{2}) = 100.
The solution involves four steps.
Therefore, the probability that the survey
will show a greater percentage of Republican voters in the
second state than in the first state is 0.24.
Note: Some analysts might have used the t-distribution to compute probabilities
for this problem. We chose the normal distribution because the population variance was known
and the sample size was large. But it would not have been wrong to use the t-distribution. In a previous lesson, we offered some guidelines for
choosing between the normal and the t-distribution.