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

7.4.3.5.

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 ± 1.663.

That is, the confidence interval is (1.557, 4.883).

Additional 95 % confidence intervals

A 95 % confidence interval for \(\mu_3 - \mu_2\) is: (-0.823, 2.503).

A 95 % confidence interval for \(\mu_2 - \mu_1\) is: (0.717, 4.043).

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.