• Check the assumptions for a two-sample [latex]t[/latex] confidence interval for population mean.
• Calculate and explain a confidence interval for the difference between two population means.

Confidence Interval for Difference in Population Means

In previous lessons, you constructed confidence interval estimates for a population proportion, a difference in proportions, and a population mean.

The form of those confidence intervals was:

estimate [latex]\pm[/latex] margin of error

When you are interested in estimating a difference in population means using data from independent samples, the confidence interval has the same form. The estimate used to construct the interval is the difference in sample means, [latex]\bar{x}_1 - \bar{x}_2[/latex], and the margin of error is calculated using the standard error for a difference in sample means and a critical value from the [latex]t[/latex]-distribution.

Compared to the formula for proportions, the margin of error here is calculated a little differently: Instead of multiplying the value of the standard error by a value from the normal distribution, it is multiplied by a value from the appropriate [latex]t[/latex]-distribution. This is not surprising if you think back to your work with the standardized [latex]t[/latex]-statistic.

Margin of Error = ([latex]t[/latex]-critical)(standard error) = ([latex]t[/latex]-critical)[latex](\sqrt{\frac{{{s}_{1}}^{2}}{{n}_{1}}+\frac{{{s}_{2}}^{2}}{{n}_{2}}})[/latex]

We will use technology to calculate the margin of error, so we won’t worry about remembering the formula for standard error or margin of error at this point.