Statistics
Variance of Observations
Grade 11
Question:
<p>The mean and variance of 8 observations are 10 and 13.5 respectively. If 6 of these observations are 5, 7, 10, 12, 14, 15, then the absolute difference of the remaining two observations is</p>
<p>(a) 9</p>
<p>(b) 3</p>
<p>(c) 7</p>
<p>(d) 5</p>
Step-by-Step Solution
Key Concept: Use the mean and variance formulas together with the sum and sum of squares of known observations to find the unknown observations.
<p><strong>Step 1:</strong> Let the remaining two observations be \(x\) and \(y\).</p><p><strong>Step 2:</strong> Using the mean condition:</p><p>\[\frac{5 + 7 + 10 + 12 + 14 + 15 + x + y}{8} = 10\]</p><p>\[63 + x + y = 80\]</p><p>\[x + y = 17 \quad \text{...(i)}\]</p><p><strong>Step 3:</strong> Using the variance condition:</p><p>\[\text{Variance} = \frac{25 + 49 + 100 + 144 + 196 + 225 + x^2 + y^2}{8} - 100 = 13.5\]</p><p>\[\frac{739 + x^2 + y^2}{8} - 100 = 13.5\]</p><p>\[739 + x^2 + y^2 = 908\]</p><p>\[x^2 + y^2 = 169\]</p><p><strong>Step 4:</strong> From equation (i): \((x+y)^2 = 289\)</p><p>\[x^2 + y^2 + 2xy = 289\]</p><p>\[169 + 2xy = 289\]</p><p>\[xy = 60\]</p><p><strong>Step 5:</strong> Finding \(|x - y|\):</p><p>\[(x-y)^2 = (x+y)^2 - 4xy = 289 - 240 = 49\]</p><p>\[|x - y| = 7\]</p><p>∴ Answer is (c) 7.</p>
Correct Answer: C