Tests of Linear Combinations of Means, Independent Groups (2 of 7)

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The steps for testing a linear contrast follow:

1. 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_iµ_i is equal to some specified value, usually zero. In this experiment, a₁ = 0.5, a₂ = 0.5, and a₃ = -1. The null hypothesis is Σa_iµ_i = 0 which can be written as: a₁µ₁ + a₂µ₂ + a₃ µ₃ = 0. For this experiment, the null hypothesis is:
   
   (0.5)(μ_aspirin) + (0.5)(μ_tylenol) + (-1)(μ_placebo) = 0.
   
   The alternative hypothesis is:
   a₁µ₁ + a₂µ₂ + a₃ µ₃ ≠ 0.
   
   
2. The second step is to choose a significance level. Assume the .05 level is chosen.
   
   
3. The third step is to compute the value of the linear combination (L) based on the samples.
   
   L = a₁M₁ + a₂M₂ + ... + a_a M_a
   
   For these data,
   
   L = (0.5)(4) + (0.5)(3) + (-1)(2) = 1.5.