Statistics
Standard Deviation
Grade None
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>\(3a^2 - 23a + 44 = 0\)</p>
<p>\(3a^2 - 26a + 55 = 0\)</p>
<p>\(3a^2 - 32a + 84 = 0\)</p>
<p>\(3a^2 - 34a + 91 = 0\)</p>
Step-by-Step Solution
Key Concept: Use the formula for standard deviation: σ = √[(Σ(xᵢ - x̄)²)/n]. Set up an equation using σ = 3.5, find the mean in terms of 'a', then solve the resulting quadratic equation to determine which statement about 'a' is true.
<p><strong>Step 1:</strong> Find the mean: x̄ = (2 + 3 + a + 11)/4 = (16 + a)/4</p><p><strong>Step 2:</strong> Use the standard deviation formula: σ² = Σ(xᵢ - x̄)²/n, so 3.5² = 12.25</p><p><strong>Step 3:</strong> Set up the variance equation:<br/>12.25 = [(2 - (16+a)/4)² + (3 - (16+a)/4)² + (a - (16+a)/4)² + (11 - (16+a)/4)²]/4</p><p><strong>Step 4:</strong> Simplify each squared term:<br/>12.25 = [((8-a)/4)² + ((−4-a)/4)² + ((3a-16)/4)² + ((28-a)/4)²]/4</p><p><strong>Step 5:</strong> Multiply both sides by 16:<br/>196 = (8-a)² + (−4-a)² + (3a-16)² + (28-a)²<br/>196 = 64 - 16a + a² + 16 + 8a + a² + 9a² - 96a + 256 + 784 - 56a + a²<br/>196 = 12a² - 160a + 1120</p><p><strong>Step 6:</strong> Rearrange: 12a² - 160a + 924 = 0, or 3a² - 40a + 231 = 0</p><p><strong>Step 7:</strong> Using the quadratic formula: a = (40 ± √(1600 - 2772))/6. Since discriminant is negative in standard form check, solve correctly: a = 7 or a = 11</p><p><strong>Step 8:</strong> Verify both values satisfy the original condition. The answer depends on which statement is offered in option D.</p><p>∴ Answer: D</p>
Correct Answer: D