10 - Inference About Means and Proportions with Two Populations

10.1 Inferences About the Difference Between Two Population Means: \(\sigma_1\) and \(\sigma_2\) Known

Using \(\mu_1\) and \(\mu_2\) to denote the population means of two populations, we can make inferences about the difference between these means: \(\mu_1 - \mu_2\). To make these inferences, we must take independent random samples with sizes of \(n_1\) and \(n_2\), respectively. In this section, we assume that the population standard deviations \(\sigma_1\) and \(\sigma_2\) are already known.

Interval Estimation of \(\mu_1 - \mu_2\)

To find this interval, we must first establish the point estimator for the difference between these two population means as \(\bar{x}_1 - \bar{x}_2\). We can then also calculate the standard error of \(\bar{x}_1 - \bar{x}_2\) from the known population standard deviations:

\[ \sigma_{\bar{x}_1 - \bar{x}_2} = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}} \]

Then, we must calculate the margin of error using this value:

\[ \text{Margin of error } = z_{\alpha/2} \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}} \]

Thus, the interval estimate for the difference between two population means where both population standard deviations are known is given by:

\[ \bar{x}_1 - \bar{x}_2 \pm z_{\alpha/2} \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}} \]

where \(1-\alpha\) is the confidence coefficient.

Hypothesis Tests About \(\mu_1 - \mu_2\)

The general process for hypothesis testing remains the same. However, the test statistic is calculated as such:

\[ z = \frac{(x_1 - x_2) - D_0}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}} \]

where \(D_0\) is the assumed difference in population means (typically \(0\)).

10.2 Inferences About the Difference Between Two Population Means: \(\sigma_1\) and \(\sigma_2\) Unknown

With the population standard deviations unknown, we must use the \(t\) distribution rather than the normal distribution to estimate the difference in population means.

Interval Estimate of \(\mu_1 - \mu_2\)

When creating an interval estimate of the difference between population means when the population standard deviation is unknown, we use a similar equation to when it is known:

\[ \bar{x}_1 - \bar{x}_2 \pm t_{\alpha/2} \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \]

where \(1-\alpha\) is the confidence coefficient. The degrees of freedom are calculated using the following formula:

\[ df = \frac{\left( \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} \right)^2}{\frac{1}{n_1 - 1} \left( \frac{s_1^2}{n_1} \right)^2 + \frac{1}{n_2-1} \left( \frac{s_2^2}{n_2} \right)^2} \]

Note that this values is typically rounded down, since that is the more conservative estimate.

Hypothesis Tests About \(\mu_1 - \mu_2\)

We can similarly adapt the previous equation for the test statistic to use the estimated standard deviation:

\[ t = \frac{(x_1 - x_2) - D_0}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \]

We then calculate the degrees of freedom using the same formula from the previous section. Using this, we can calculate a \(p\)-value for the hypothesis test.

10.3 Inferences About the Difference Between Two Population Means: Matched Samples

An alternative to independent sampling is matched sampling, which uses the same sample for both treatments (ex. same employees performing two different methods). Since this method helps minimize sampling variability, it leads to a smaller sampling error.

When analyzing the difference in means, we must first calculate the sample mean and sample standard deviation of the differences \(d\).

\[ \bar{d} = \frac{\sum d_i}{n} \]

\[ s_d = \sqrt{\frac{\sum (d_i - \bar{d})^2}{n-1}} \]

With this, we can calculate the test statistic for analyzing this difference in means:

\[ t = \frac{\bar{d} - \mu_d}{s_d / \sqrt{n}} \]

This test statistic has a \(t\) distribution with \(n-1\) degrees of freedom. This test statistic can then be used to create confidence intervals, calculate \(p\)-values, or otherwise perform a hypothesis test.

10.4 Inferences About the Difference Between Two Population Proportions

Let \(p_1 - p_2\) denote the difference between the population proportions of two populations. We will then draw a sample of size \(n_1\) from population \(1\) and a sample of size \(n_2\) from population \(2\).

Interval Estimation of \(p_1 - p_2\)

The point estimator for the difference in population proportions is \(\bar{p}_1 - \bar{p}_2\). The standard error of this distribution can be calculated as such:

\[ \sigma_{\bar{p}_1 - \bar{p}_2} = \sqrt{\frac{p_1(1-n_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}} \]

If the sample size is large enough such that \(n_1 p_1\), \(n_1 (1-p_1)\), \(n_2 p_2\), and \(n_2 (1-p_2)\) are all less than \(5\), then the sampling distribution of \(\bar{p}_1 - \bar{p}_2\) can be modeled by the normal distribution.

To perform an interval estimation, we must first calculate the margin of error:

\[ \text{Margin of error } = z_{\alpha/2} \sqrt{\frac{p_1(1-n_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}} \]

It follows that an interval estimate would take the following form:

\[ \bar{p}_1 - \bar{p}_2 \pm z_{\alpha/2} \sqrt{\frac{p_1(1-n_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}} \]

Hypothesis Tests About \(p_1 - p_2\)

In a hypothesis test, we typically have the null hypothesis be an assumption that the difference of population proportions is related to \(0\) in some way. Note that this is the same as saying that \(p_1 = p_2\), so we can simplify our equation for the standard error \(\sigma_{\bar{p}_1 - \bar{p}_2}\).

\[ \sigma_{\bar{p}_1 - \bar{p}_2} = \sqrt{p(1-p) \left( \frac{1}{n_1}+\frac{1}{n_2} \right)} \]

We then also establish a pooled estimator of \(p\), which is a weighted average of \(\bar{p}_1\) and \(\bar{p}_2\):

\[ \bar{p} = \frac{n_1 \bar{p}_1 + n_2 \bar{p}_2}{n_1 + n_2} \]

With this, we can calculate the test statistic for this hypothesis test:

\[ z = \frac{\bar{p}_1 - \bar{p}_2}{\sqrt{p(1-p) \left( \frac{1}{n_1}+\frac{1}{n_2} \right)}} \]

This test statistic applies to large samples where \(n_1 p_1\), \(n1 (1-p_1)\), \(n_2 p_2\), and \(n_2 (1-p_2)\) are greater or equal to \(5\).