<p>The mean of 100 observations is 50 and their standard deviation is 5. The sum of squares of all the observations is</p>
Step-by-Step Solution
Key Concept: Use the fundamental relationship between mean, standard deviation, and sum of squares: Var(X) = E(X²) - [E(X)]², which directly gives us E(X²) = σ² + μ². Then multiply by n to find the sum of squares.
<p><strong>Step 1:</strong> Recall the variance formula: σ² = (Σx²/n) - μ²</p><p><strong>Step 2:</strong> Rearrange to find Σx²: Σx² = n(σ² + μ²)</p><p><strong>Step 3:</strong> Substitute n = 100, μ = 50, σ = 5</p><p>Σx² = 100(5² + 50²) = 100(25 + 2500) = 100(2525) = 252,500</p><p><strong>Step 4:</strong> Verify: Mean of squares = 252,500/100 = 2525 ✓, and 2525 - 2500 = 25 = σ² ✓</p><p>∴ Answer: C (252,500)</p>
Correct Answer: C