Statistics
Mean and Variance
Grade 11
Question:
<p>The mean of five observations is 4 and their variance is 5.2. If three of these observations are 1, 2 and 6, then the other two are</p>
<p>2 and 9</p>
<p>3 and 8</p>
<p>4 and 7</p>
<p>5 and 6</p>
Step-by-Step Solution
Key Concept: Use the mean equation to get one constraint on the two unknowns, then use the variance formula to get a second constraint. Solve the resulting quadratic system.
<p><strong>Step 1:</strong> Let the two unknown observations be <em>x</em> and <em>y</em>.</p><p>From mean = 4: (1 + 2 + 6 + x + y)/5 = 4</p><p>Therefore: 9 + x + y = 20 ⟹ <strong>x + y = 11</strong> ... (1)</p><p><strong>Step 2:</strong> Variance formula: σ² = (Σx²)/n - (mean)²</p><p>5.2 = (1² + 2² + 6² + x² + y²)/5 - 16</p><p>5.2 = (1 + 4 + 36 + x² + y²)/5 - 16</p><p>5.2 + 16 = (41 + x² + y²)/5</p><p>21.2 × 5 = 41 + x² + y²</p><p>106 = 41 + x² + y² ⟹ <strong>x² + y² = 65</strong> ... (2)</p><p><strong>Step 3:</strong> From (1): (x + y)² = 121 ⟹ x² + 2xy + y² = 121</p><p>Substituting x² + y² = 65: 65 + 2xy = 121 ⟹ 2xy = 56 ⟹ xy = 28</p><p><strong>Step 4:</strong> <em>x</em> and <em>y</em> are roots of: t² - 11t + 28 = 0</p><p>Using quadratic formula: t = (11 ± √(121 - 112))/2 = (11 ± 3)/2</p><p>Therefore: <strong>t = 7 or t = 4</strong></p><p>∴ The other two observations are <strong>4 and 7</strong></p>
Correct Answer: C