7. Product and Process Comparisons
7.4. Comparisons based on data from more than two processes
7.4.3. Are the means equal?

## Confidence intervals for the difference of treatment means

Confidence intervals for the difference between two means This page shows how to construct a confidence interval around $$(\mu_i - \mu_j)$$ for the one-way ANOVA by continuing the example shown on a previous page.
Formula for the confidence interval The formula for a $$100(1-\alpha)$$ % confidence interval for the difference between two treatment means is: $$(\hat{\mu_i} - \hat{\mu_j}) \pm t_{1-\alpha/2, \, N-k} \,\,\sqrt{\hat{\sigma}^2_\epsilon \left( \frac{1}{n_i}+\frac{1}{n_j}\right)} \, ,$$ where $$\hat{\sigma}_\epsilon^2 = MSE$$.
Computation of the confidence interval for $$\mu_3 - \mu_1$$ For the example, we have the following quantities for the formula.
• $$\bar{y}_3 = 8.56$$

• $$\bar{y}_1 = 5.34$$

• $$\sqrt{1.454(1/5 + 1/5)} = 0.763$$

• $$t_{0.975, \, 12} = 2.179$$

Substituting these values yields (8.56 - 5.34) ± 2.179(0.763) or 3.22 ± 1.616.

That is, the confidence interval is (1.604, 4.836).

Additional 95 % confidence intervals A 95 % confidence interval for $$\mu_3 - \mu_2$$ is: (-1.787, 3.467).

A 95 % confidence interval for $$\mu_2 - \mu_1$$ is: (-0.247, 5.007).

Contrasts discussed later Later on the topic of estimating more general linear combinations of means (primarily contrasts) will be discussed, including how to put confidence bounds around contrasts.