<p>The mean of the numbers \(a, b, 8, 5\), and 10 is 6 and the variance is 6.80. Then which one of the following gives the possible values of \(a\) and \(b\)?</p>
Step-by-Step Solution
Key Concept: Use the mean condition to establish a linear relationship between a and b, then substitute into the variance formula to get a quadratic constraint. This simultaneous system yields the possible values.
<p><strong>Step 1: Use the mean condition</strong></p><p>Mean of a, b, 8, 5, 10 is 6:</p><p>(a + b + 8 + 5 + 10)/5 = 6</p><p>a + b + 23 = 30</p><p>a + b = 7</p><p><strong>Step 2: Apply the variance formula</strong></p><p>Variance = Σ(xᵢ - mean)²/n = 6.80</p><p>[(a-6)² + (b-6)² + (8-6)² + (5-6)² + (10-6)²]/5 = 6.80</p><p>(a-6)² + (b-6)² + 4 + 1 + 16 = 34</p><p>(a-6)² + (b-6)² = 13</p><p><strong>Step 3: Expand using b = 7 - a</strong></p><p>(a-6)² + (7-a-6)² = 13</p><p>(a-6)² + (1-a)² = 13</p><p>a² - 12a + 36 + 1 - 2a + a² = 13</p><p>2a² - 14a + 37 = 13</p><p>2a² - 14a + 24 = 0</p><p>a² - 7a + 12 = 0</p><p>(a-3)(a-4) = 0</p><p><strong>Step 4: Find the pair</strong></p><p>If a = 3, then b = 4</p><p>If a = 4, then b = 3</p><p>∴ Answer: D (a = 3, b = 4 or a = 4, b = 3)</p>
Correct Answer: D