Statistics
Variance
Grade 11
Question:
<p>The mean of the numbers \(a, b, 8, 5, 10\) is 6 and the variance is 6.80. Then which one of the following gives possible values of \(a\) and \(b\)?</p>
<p>\(a = 0,\ b = 7\)</p>
<p>\(a = 5,\ b = 2\)</p>
<p>\(a = 1,\ b = 6\)</p>
<p>\(a = 3,\ b = 4\)</p>
Step-by-Step Solution
Key Concept: Use the mean condition to get one equation, then use the variance formula (which involves the sum of squared deviations from mean) to get a second equation. Solve the system to find a and b.
<p><strong>Step 1:</strong> Use the mean condition.</p><p>Mean = (a + b + 8 + 5 + 10)/5 = 6</p><p>Therefore: a + b + 23 = 30</p><p>∴ a + b = 7</p><p><strong>Step 2:</strong> Use the variance formula.</p><p>Variance = E(X²) - [E(X)]²</p><p>First find E(X²) = (a² + b² + 64 + 25 + 100)/5 = (a² + b² + 189)/5</p><p>Given: Variance = 6.80, so:</p><p>(a² + b² + 189)/5 - 36 = 6.80</p><p>a² + b² + 189 = 5(42.80) = 214</p><p>∴ a² + b² = 25</p><p><strong>Step 3:</strong> Solve the system.</p><p>From a + b = 7: b = 7 - a</p><p>Substitute into a² + b² = 25:</p><p>a² + (7-a)² = 25</p><p>a² + 49 - 14a + a² = 25</p><p>2a² - 14a + 24 = 0</p><p>a² - 7a + 12 = 0</p><p>(a - 3)(a - 4) = 0</p><p>∴ a = 3, b = 4 or a = 4, b = 3</p><p><strong>Verification:</strong> Sum = 3 + 4 = 7 ✓; Sum of squares = 9 + 16 = 25 ✓</p><p>∴ Answer: D (the option containing {3,4} or {4,3})</p>
Correct Answer: D