<p>If each observation of a raw data whose variance is \(\sigma\) is multiplied by \(h\), then the variance of the new set is</p>
Step-by-Step Solution
Key Concept: Variance scales with the square of the scaling factor: when each observation is multiplied by h, the variance (which measures squared deviations from mean) gets multiplied by h². This follows from the property that Var(kX) = k²Var(X).
<p><strong>Step 1:</strong> Recall the definition of variance: σ² = (1/n)Σ(xᵢ - x̄)²</p><p><strong>Step 2:</strong> When each observation xᵢ is multiplied by h, the new observations become hxᵢ. The new mean becomes h·x̄ (mean also scales linearly).</p><p><strong>Step 3:</strong> New variance = (1/n)Σ(hxᵢ - h·x̄)² = (1/n)Σ[h(xᵢ - x̄)]² = (1/n)Σh²(xᵢ - x̄)² = h² · (1/n)Σ(xᵢ - x̄)²</p><p><strong>Step 4:</strong> New variance = h² · σ (original variance becomes h² times the original)</p><p>∴ Answer: <strong>h²σ</strong> (Option B)</p>
Correct Answer: B