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: Σaiµi is equal to some specified value, usually zero. In this experiment, a1 = 0.5, a2 = 0.5, and a3 = -1. The null hypothesis is Σaiµi = 0 which can be written as: a1µ1 + a2µ2 + a3 µ3 = 0. For this experiment, the null hypothesis is:

    (0.5)(μaspirin) + (0.5)(μtylenol) + (-1)(μplacebo) = 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 + ... + aa Ma

    For these data,

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