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For 18 observations X₁, X₂,..., X₁₈, it is given that sum(X_i - alpha) = 36 and sum(X_i - beta)² = 90, where alpha and beta are distinct real numbers. If the standard deviation of the observations is 1, find |alpha - beta|.

1. (A) 1
2. (B) 2
3. (C) 3
4. (D) 4

Correct answer: (D) 4

Solution

From sum(X_i - alpha) = 36 and n=18, the mean x-bar = alpha + 2. The variance identity gives 90/18 = 1 + (x-bar - beta)², so (x-bar - beta)² = 4, meaning |x-bar - beta| = 2. Since x-bar = alpha + 2, we have |alpha + 2 - beta| = 2. Since alpha != beta the only non-trivial solution is alpha - beta = -4, giving |alpha - beta| = 4.

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