Compute the average effect of the interaction between X1 and X3.

Question

Options

• A-2
• B+1/2
• C+3
• D1/3
• Enone of the above

• A
  9% (3)
• B
  61% (20)
• C
  21% (7)
• D
  6% (2)
• E
  3% (1)

Explanation

Option B (+1/2) is correct because the average interaction effect between X1 and X3 is computed as half the interaction contrast - that is, the sum of responses where X1 and X3 have the same sign (both + or both −) minus those with opposite signs, divided by the appropriate number of observations. Applying this formula to the given data yields a contrast of +1, and dividing by 2 (to get the average effect) gives +1/2.

Option A (−2) is a sign error combined with a scaling mistake, likely from misassigning the signs in the X1X3 contrast column. Option C (+3) comes from summing the relevant column without dividing to obtain the average. Option D (1/3) reflects dividing by the wrong number - using 3 observations instead of the correct divisor for the design. All three distractors reflect the most common arithmetic traps: forgetting to halve the contrast, or dividing by the wrong count.

Memory tip: For any 2-level factorial interaction, the formula is (same-sign total − opposite-sign total) ÷ (n/2). If you get a whole number, suspect you forgot to divide - the average effect always involves that extra halving step.

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