Statistics
Standard Deviation
Grade 11
Question:
<p>If the standard deviation of the numbers 2, 3, \(a\) and 11 is 3.5, then which of the following is true?</p>
<p>\(3a^2 - 32a + 84 = 0\)</p>
<p>\(3a^2 - 34a + 91 = 0\)</p>
<p>\(3a^2 - 23a + 44 = 0\)</p>
<p>\(3a^2 - 26a + 55 = 0\)</p>
Step-by-Step Solution
Key Concept: Use the variance formula σ² = (Σx²/n) - (mean)² to create an equation in terms of 'a', then solve for the possible values that satisfy σ = 3.5.
<p><strong>Step 1:</strong> Calculate the mean: μ = (2 + 3 + a + 11)/4 = (16 + a)/4</p><p><strong>Step 2:</strong> Use the variance formula σ² = Σx²/n - μ²</p><p>Here, σ = 3.5, so σ² = 12.25</p><p>Σx² = 4 + 9 + a² + 121 = 134 + a²</p><p><strong>Step 3:</strong> Set up the equation: 12.25 = (134 + a²)/4 - [(16 + a)/4]²</p><p>12.25 = (134 + a²)/4 - (16 + a)²/16</p><p><strong>Step 4:</strong> Multiply by 16: 196 = 4(134 + a²) - (16 + a)²</p><p>196 = 536 + 4a² - (256 + 32a + a²)</p><p>196 = 536 + 4a² - 256 - 32a - a²</p><p>196 = 280 + 3a² - 32a</p><p>3a² - 32a + 84 = 0</p><p><strong>Step 5:</strong> Using the quadratic formula: a = (32 ± √(1024 - 1008))/6 = (32 ± 4)/6</p><p>a = 6 or a = 14/3</p><p>∴ Answer: A</p>
Correct Answer: A