<p>For two data sets, each of size 5, the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4, respectively. The variance of the combined data set is</p>
Step-by-Step Solution
Key Concept: The variance of combined data sets requires combining both the internal variances and the variance arising from differences between individual means and the overall mean. Use the formula: σ²_combined = [n₁σ₁² + n₂σ₂² + n₁(m₁-M)² + n₂(m₂-M)²]/(n₁+n₂), where M is the combined mean.
<p><strong>Step 1:</strong> Find the combined mean M of both datasets.</p><p>M = (n₁m₁ + n₂m₂)/(n₁ + n₂) = (5×2 + 5×4)/(5+5) = (10+20)/10 = 3</p><p><strong>Step 2:</strong> Apply the combined variance formula.</p><p>σ²_combined = [n₁σ₁² + n₂σ₂² + n₁(m₁-M)² + n₂(m₂-M)²]/(n₁+n₂)</p><p><strong>Step 3:</strong> Substitute values: σ₁² = 4, σ₂² = 5, m₁ = 2, m₂ = 4, M = 3, n₁ = n₂ = 5</p><p>σ²_combined = [5(4) + 5(5) + 5(2-3)² + 5(4-3)²]/10</p><p>σ²_combined = [20 + 25 + 5(1) + 5(1)]/10 = [20 + 25 + 5 + 5]/10 = 55/10 = 5.5</p><p><strong>∴ Answer: C (5.5)</strong></p>
Correct Answer: C