The steps for testing a linear contrast follow:

- The first step is
to specify the null hypothesis and
an alternative hypothesis. For
experiments testing linear combinations of means, the null
hypothesis is: Σa
_{i}µ_{i }is equal to some specified value, usually zero. In this experiment, a_{1}= 0.5, a_{2}= 0.5, and a_{3}= -1. The null hypothesis is Σa_{i}µ_{i}= 0 which can be written as: a_{1}µ_{1}+ a_{2}µ_{2}+ a_{3}µ_{3}= 0. For this experiment, the null hypothesis is:

(0.5)(μ_{aspirin}) + (0.5)(μ_{tylenol}) + (-1)(μ_{placebo}) = 0.

The alternative hypothesis is:

a_{1}µ_{1}+ a_{2}µ_{2}+ a_{3}µ_{3}≠ 0.

- The second step is to
choose a significance level. Assume the .05 level
is chosen.

- The third step is to
compute the value of the linear combination (L) based on the
samples.

L = a_{1}M_{1}+ a_{2}M_{2}+ ... + a_{a}M_{a }For these data,

L = (0.5)(4) + (0.5)(3) + (-1)(2) = 1.5.