Module 10: Inference for Means
Estimating the Difference in Two Population Means
Estimating the Difference in Two Population Means
Learning outcomes
- Construct a confidence interval to estimate a difference in two population means (when conditions are met). Interpret the confidence interval in context.
Confidence Interval to Estimate μ1 − μ2
In a hypothesis test, when the sample evidence leads us to reject the null hypothesis, we conclude that the population means differ or that one is larger than the other. An obvious next question is how much larger? In practice, when the sample mean difference is statistically significant, our next step is often to calculate a confidence interval to estimate the size of the population mean difference.
The confidence interval gives us a range of reasonable values for the difference in population means μ1 − μ2. We call this the two-sample T-interval or the confidence interval to estimate a difference in two population means. The form of the confidence interval is similar to others we have seen.
(samplestatistic)±(marginoferror)
(samplestatistic)±(criticalT−value)(standarderror)
Sample Statistic
Since we’re estimating the difference between two population means, the sample statistic is the difference between the means of the two independent samples: [latex]\bar{x}_{1}-\bar{x}_{2}[/latex].
Critical T-Value
The critical T-value comes from the T-model, just as it did in “Estimating a Population Mean.” Again, this value depends on the degrees of freedom (df). For two-sample T-test or two-sample T-intervals, the df value is based on a complicated formula that we do not cover in this course. We either give the df or use technology to find the df.
Standard Error
The estimated standard error for the two-sample T-interval is the same formula we used for the two-sample T-test. (As usual, s1 and s2 denote the sample standard deviations, and n1 and n2 denote the sample sizes.)
[latex]\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}[/latex]
Putting all this together gives us the following formula for the two-sample T-interval.
[latex](\bar{x_{1}}-\bar{x_{2}})\pm T_{c}\cdot \sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}[/latex]
Conditions for Use
The conditions for using this two-sample T-interval are the same as the conditions for using the two-sample T-test.
- The two random samples are independent and representative.
- The variable is normally distributed in both populations. If it is not known, samples of more than 30 will have a difference in sample means that can be modeled adequately by the T-distribution. As we discussed in “Hypothesis Test for a Population Mean,” T-procedures are robust even when the variable is not normally distributed in the population. If checking normality in the populations is impossible, then we look at the distribution in the samples. If a histogram or dotplot of the data does not show extreme skew or outliers, we take it as a sign that the variable is not heavily skewed in the p