<p>The mean and the variance of five observations are 4 and 5.20, respectively. If three of the observations are 3, 4 and 4; then the absolute value of the difference of the other two observations, is:</p>
Step-by-Step Solution
Key Concept: Use the definitions of mean and variance to set up two equations in two unknowns (the unknown observations), then solve the resulting quadratic to find both unknown values.
<p><strong>Step 1:</strong> Let the five observations be 3, 4, 4, x, y where x and y are unknown.</p><p><strong>Step 2:</strong> From mean = 4: (3 + 4 + 4 + x + y)/5 = 4 → x + y = 9</p><p><strong>Step 3:</strong> From variance = 5.20: E(x²) - [E(x)]² = 5.20 → E(x²) = 5.20 + 16 = 21.20</p><p><strong>Step 4:</strong> Therefore: (9 + 16 + 16 + x² + y²)/5 = 21.20 → x² + y² = 106 - 41 = 65</p><p><strong>Step 5:</strong> We have x + y = 9 and x² + y² = 65. Using (x + y)² = x² + y² + 2xy: 81 = 65 + 2xy → xy = 8</p><p><strong>Step 6:</strong> So x and y are roots of t² - 9t + 8 = 0 → (t - 1)(t - 8) = 0 → t = 1 or t = 8</p><p><strong>Step 7:</strong> The two observations are 1 and 8, so |x - y| = |8 - 1| = 7</p><p>∴ Answer: 7 (Option D)</p>
Correct Answer: D