<p>The mean of five observations is 5 and their variance is 9.20. If three of the given five observations are 1, 3 and 8, then a ratio of other two observations is:</p>
Step-by-Step Solution
Key Concept: Use the mean and variance formulas to set up two equations with two unknowns (the unknown observations), then solve the resulting quadratic to find the pair of values and their ratio.
<p><strong>Step 1:</strong> Let the five observations be 1, 3, 8, x, y where x and y are unknown.</p><p><strong>Step 2:</strong> From mean = 5: (1 + 3 + 8 + x + y)/5 = 5 → x + y = 13</p><p><strong>Step 3:</strong> From variance = 9.20, use Var = Σxᵢ²/n - (x̄)²</p><p>Variance = (1 + 9 + 64 + x² + y²)/5 - 25 = 9.20</p><p>(74 + x² + y²)/5 = 34.20</p><p>74 + x² + y² = 171</p><p>x² + y² = 97</p><p><strong>Step 4:</strong> From x + y = 13, we get (x + y)² = 169</p><p>x² + y² + 2xy = 169</p><p>97 + 2xy = 169</p><p>xy = 36</p><p><strong>Step 5:</strong> x and y are roots of t² - 13t + 36 = 0</p><p>(t - 4)(t - 9) = 0</p><p>∴ x = 4, y = 9 (or vice versa)</p><p><strong>Step 6:</strong> Ratio of the two observations = 4:9</p><p>∴ Answer: D</p>
Correct Answer: D