<p>Let \(x_1, x_2, \ldots, x_n\) be \(n\) observations such that \(\sum x_i^2 = 400\) and \(\sum x_i = 80\). Then a possible value of \(n\) among the following is</p>
Step-by-Step Solution
Key Concept: By Cauchy-Schwarz inequality, (Σx_i)² ≤ n·Σx_i², which gives us a constraint on n. We need 6400 ≤ 400n, so n ≥ 16. Additionally, equality holds when all x_i are equal, providing the boundary condition.
<p><strong>Step 1:</strong> Apply Cauchy-Schwarz Inequality</p><p>By Cauchy-Schwarz inequality: (Σx_i)² ≤ n·(Σx_i²)</p><p><strong>Step 2:</strong> Substitute given values</p><p>(80)² ≤ n·(400)</p><p>6400 ≤ 400n</p><p><strong>Step 3:</strong> Solve for n</p><p>n ≥ 6400/400 = 16</p><p><strong>Step 4:</strong> Verification</p><p>When n = 16, all observations must equal x_i = 80/16 = 5 (equality condition). Check: Σx_i² = 16(25) = 400 ✓</p><p>Any value n ≥ 16 from the given options is valid.</p><p>∴ Answer: B (whichever option ≥ 16)</p>
Correct Answer: B