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 :

Solution

Maths sir · 3 steps

Mean gives you a + b, variance gives you a² + b², and together they hand you (a − b)².

1. Use the mean. Call the two missing observations a and b. The total of all 8 is 8 × 10 = 80, and the six known values add to 5+7+10+12+14+15 = 63. So a + b = 17.
2. Use the variance. From σ² = (Σ x²)/n - x̄², we get Σ x² = 8 × (13.5 + 100) = 908. The known squares add to 25+49+100+144+196+225 = 739, so a² + b² = 169.

   (a-b)² = 2(a²+b²) - (a+b)² = 338 - 289 = 49

3. So. |a - b| = 7. The two observations are 12 and 5, and 12 + 5 = 17 with 144 + 25 = 169 confirms it. The answer is (c).

The idea to keep. With two unknown observations, the mean fixes their sum and the variance fixes their sum of squares. Then (a-b)² = 2(a²+b²) - (a+b)² gives the difference without solving for each.

Answer from the NTA final answer key. Solution by PrepWiser.