### Differences in Two Population Means

For a confidence interval on the difference in two population means to be valid, two conditions must be met. First, samples must be selected randomly and independently from the two populations. Second, for each population, the responses must be normally distributed or the sample size must be sufficiently large to invoke the Central Limit Theorem. If these two conditions are satisfied, the process of establishing the confidence intervals is the same as that used for a two group design.

#### Example

Set a 95% confidence interval on the difference in the mean difference in heights for fraternal and identical twins using the data in the previous lesson.

#### Solution

First, consider the conditions. The twins were recruited and not randomly selected from all fraternal and identical twins. We must assume that these samples are representative of the populations if we are to proceed. We will make this assumption, knowing that it is a potential weakness in our study. Second, only nonnegative values can be observed, and the sample distributions appear skewed. Therefore, the distribution of differences in twin heights is not normal for either fraternal or identical twins. However, the sample size is 30, so we will assume that the Central Limit Theorem can be applied.

The estimated mean difference in the heights for fraternal and identical twins is – = 2.4 cm, and the standard error of this estimate is _{,} where the subscript *F* indicates fraternal twins and the subscript *I* represents identical twins. The degrees of freedom are

.

To look up the tabulated value, we will round to the nearest integer, 37 in this case. In the row corresponding to 37 *df* and the column under 0.025 in the *t*-table, we have 2.024. Based on these values, the 95% confidence interval for μ_{F} – μ_{I} is 2.4 ± 2.024 or 2.4 ± 1.518. We estimate that the mean difference in fraternal twins' heights is 2.4 cm greater than the mean difference in identical twins' heights, and we are 95% confident this estimate is within 1.5 cm of the difference in these two population means.

### Differences in Two Population Proportions

Sometimes, we want to estimate the difference in two population proportions. For example, we want to estimate "the gender gap," or the difference in the proportions of men and women favoring a particular candidate. Two conditions must be satisfied to use the methods discussed here. First, independent samples are randomly selected from each of the populations. Suppose *n*_{1}and *n*_{2} are the number of observations from populations 1 and 2, respectively. Further, the sample proportions, 1 and 2, respectively. Further, the sample proportions, and , are the estimates of population 1 and 2 proportions, *p*_{1} – *p*_{2}, respectively. The second condition is that *n*_{1}, *n*_{1}(1 – ), *n*_{2}, and *n*_{2}(1 – ) are all at least 5, and preferably at least 10.

The estimate of the difference in population proportions, *p*_{1} – *p*_{2}, is – . The standard error of this estimate is . Standardizing – , we have Therefore, the 100(1 – α) % confidence interval is (– ) ±where *z** is the tabulated value of *z* such that .

### In Short

In a two-group design, treatments are randomly assigned to the experimental units. For a two-group design, the methods for setting confidence intervals on the difference in two treatment means were discussed. An important step in this process is determining whether or not the variances of the units under each treatment are equal. The methods are the same when comparing the means of two populations or two population proportions. Although not covered here, the procedures for hypothesis testing have the same extensions as those for confidence intervals.

Find practice problems and solutions for these concepts at Confidence Intervals for Comparing Two Treatment or Population Means Practice Problems.

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