<p>Consider the first 10 positive integers. If we multiply each number by \(-1\) and then add 1 to each number, the variance of the numbers so obtained is</p>
Step-by-Step Solution
Key Concept: Variance is invariant under linear transformations of the form aX + b; only the scaling factor 'a' affects variance as Var(aX + b) = a²·Var(X). Here the transformation is -X + 1, so variance gets multiplied by (-1)² = 1.
<p><strong>Step 1:</strong> Find variance of original numbers {1, 2, 3, ..., 10}.</p><p>Mean = (1+2+...+10)/10 = 55/10 = 5.5</p><p>Variance = Σ(xᵢ - 5.5)²/10 = [(-4.5)² + (-3.5)² + ... + (4.5)²]/10</p><p>= [20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + 20.25]/10</p><p>= 82.5/10 = <strong>8.25</strong> or <strong>33/4</strong></p><p><strong>Step 2:</strong> Apply transformation rule Var(aX + b) = a²·Var(X).</p><p>New data: Y = -1·X + 1 (where a = -1, b = 1)</p><p>Var(Y) = (-1)²·Var(X) = 1·(8.25) = <strong>8.25</strong></p><p>∴ Answer: A (Variance = 8.25 or 33/4)</p>
Correct Answer: A