Letting p1 denote the proportion for population 1 and p2 denote the proportion for population 2, we next consider inferences about the difference between the two population proportions: P1 – P2. To make an inference about this difference, we will select two independent random samples consisting of n1 units from population 1 and n2 units from population 2.
1. Interval Estimation of p1 — p2
In the following example, we show how to compute a margin of error and develop an interval estimate of the difference between two population proportions.
A tax preparation firm is interested in comparing the quality of work at two of its regional offices. By randomly selecting samples of tax returns prepared at each office and verifying the sample returns’ accuracy, the firm will be able to estimate the proportion of erroneous returns prepared at each office. Of particular interest is the difference between these proportions.
The difference between the two population proportions is given by p1 – p2. The point estimator of p1 – p2 is as follows.
Thus, the point estimator of the difference between two population proportions is the difference between the sample proportions of two independent simple random samples.
As with other point estimators, the point estimator p1 – p2 has a sampling distribution that reflects the possible values of p1 – p2 if we repeatedly took two independent random samples. The mean of this sampling distribution is p1 – p2 and the standard error of p1 – p2 is as follows:
If the sample sizes are large enough that n1p1, n1(1 − p1), n2p2, and n2(1 − p2) are all greater than or equal to 5, the sampling distribution of p1 – p2 can be approximated by a normal distribution.
As we showed previously, an interval estimate is given by a point estimate ± a margin of error. In the estimation of the difference between two population proportions, an interval estimate will take the following form:
p1 – p2 ± Margin of error
With the sampling distribution of p1 — p2 approximated by a normal distribution, we would like to use za/2&pi _p as the margin of error. However, spi _p given by equation (10.11) cannot be used directly because the two population proportions, p1 and p2, are unknown. Using the sample proportion p1 to estimate p1 and the sample proportion p2 to estimate p2, the margin of error is as follows.
The general form of an interval estimate of the difference between two population proportions is as follows.
Returning to the tax preparation example, we find that independent simple random samples from the two offices provide the following information.
The sample proportions for the two offices follow.
The point estimate of the difference between the proportions of erroneous tax returns for the two populations is p1 — p2 = .14 — .09 = .05. Thus, we estimate that office 1 has a .05, or 5%, greater error rate than office 2.
Expression (10.13) can now be used to provide a margin of error and interval estimate of the difference between the two population proportions. Using a 90% confidence interval with za/2 = z.05 = 1.645, we have
Thus, the margin of error is .045, and the 90% confidence interval is .005 to .095.
2. Hypothesis Tests About p1 — p2
Let us now consider hypothesis tests about the difference between the proportions of two populations. We focus on tests involving no difference between the two population proportions. In this case, the three forms for a hypothesis test are as follows:
When we assume H0 is true as an equality, we have Pi – p2 = 0, which is the same as saying that the population proportions are equal, pi = p2.
We will base the test statistic on the sampling distribution of the point estimator pi – p2. In equation (i0.ii), we showed that the standard error of pi – p2 is given by
Under the assumption H0 is true as an equality, the population proportions are equal and pi = p2 = p. In this case, σp1-p2 becomes
With p unknown, we pool, or combine, the point estimators from the two samples (pi and p2) to obtain a single point estimator of P as follows:
This pooled estimator of p is a weighted average of pi and p2.
Substituting p forp in equation (i0.i4), we obtain an estimate of the standard error of Pi – P2. This estimate of the standard error is used in the test statistic. The general form of the test statistic for hypothesis tests about the difference between two population proportions is the point estimator divided by the estimate of σp1-p2
This test statistic applies to large sample situations where n1p1, n1(1 – p1), n2p2, and n2(1 – p2) are all greater than or equal to 5.
Let us return to the tax preparation firm example and assume that the firm wants to use a hypothesis test to determine whether the error proportions differ between the two offices. A two-tailed test is required. The null and alternative hypotheses are as follows:
If H0 is rejected, the firm can conclude that the error rates at the two offices differ. We will use a = .10 as the level of significance.
The sample data previously collected showed p1 = .14 for the n1 = 250 returns sampled at office 1 and p2 = .09 for the n2 = 300 returns sampled at office 2. We continue by computing the pooled estimate of p.
Using this pooled estimate and the difference between the sample proportions, the value of the test statistic is as follows.
In computing the p-value for this two-tailed test, we first note that z = 1.85 is in the upper tail of the standard normal distribution. Using z = 1.85 and the standard normal distribution table, we find the area in the upper tail is 1.0000 – .9678 = .0322. Doubling this area for a two-tailed test, we find the p-value = 2(.0322) = .0644. With the p-value less than a = .10, H0 is rejected at the .10 level of significance. The firm can conclude that the error rates differ between the two offices. This hypothesis testing conclusion is consistent with the earlier interval estimation results that showed the interval estimate of the difference between the population error rates at the two offices to be .005 to .095, with Office 1 having the higher error rate.
Source: Anderson David R., Sweeney Dennis J., Williams Thomas A. (2019), Statistics for Business & Economics, Cengage Learning; 14th edition.