<p>Consider the frequency distribution, where <em>A</em> is a positive integer:</p><table border='1' cellpadding='4'><tr><th>\(x\)</th><td>\(A\)</td><td>\(2A\)</td><td>\(3A\)</td><td>\(4A\)</td><td>\(5A\)</td><td>\(6A\)</td></tr><tr><th>\(f\)</th><td>2</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr></table><p>If the variance is 160, then the value of <em>A</em> is</p>
Step-by-Step Solution
Key Concept: Variance is calculated as the mean of squares minus the square of the mean. We use the formula Var(X) = E(X²) - [E(X)]² to set up an equation and solve for A.
<p><strong>Step 1: Identify the data</strong></p><p>We have values: x = A, 2A, 3A, 4A, 5A, 6A with frequencies f = 2, 1, 1, 1, 1, 1 respectively.</p><p>Total frequency: n = 2 + 1 + 1 + 1 + 1 + 1 = 7</p><p><strong>Step 2: Calculate the mean</strong></p><p>Mean = (1/n)Σ(x·f) = (1/7)[2(A) + 1(2A) + 1(3A) + 1(4A) + 1(5A) + 1(6A)]</p><p>= (1/7)[2A + 2A + 3A + 4A + 5A + 6A]</p><p>= (1/7)[22A] = 22A/7</p><p><strong>Step 3: Calculate E(X²)</strong></p><p>E(X²) = (1/n)Σ(x²·f) = (1/7)[2(A)² + 1(2A)² + 1(3A)² + 1(4A)² + 1(5A)² + 1(6A)²]</p><p>= (1/7)[2A² + 4A² + 9A² + 16A² + 25A² + 36A²]</p><p>= (1/7)[92A²] = 92A²/7</p><p><strong>Step 4: Apply variance formula</strong></p><p>Var(X) = E(X²) - [E(X)]²</p><p>160 = 92A²/7 - (22A/7)²</p><p>160 = 92A²/7 - 484A²/49</p><p><strong>Step 5: Simplify</strong></p><p>160 = (92A² × 7 - 484A²)/49</p><p>160 = (644A² - 484A²)/49</p><p>160 = 160A²/49</p><p>160 × 49 = 160A²</p><p>49 = A²</p><p>A = 7 (since A is positive)</p><p><strong>Verification:</strong> If A = 7, then Var = 160(49)/49 = 160 ✓</p><p><strong>∴ Answer: A</strong></p>
Correct Answer: A