Let the Mean and Variance of five observations x₁=1, x₂=3, x₃=a, x₄=7 and x₅=b, a>b, be 5 and 10 respectively. Then the Variance of the observations n+xₙ, n=1,2, …, 5 is

Solution

Maths sir · 5 steps

Pin down a and b from the mean and variance first, then build the new data and compute directly.

1. Use the mean. The mean is 5, so the sum is 25: 1 + 3 + a + 7 + b = 25, which gives a + b = 14.
2. Use the variance. Variance = (Σ xᵢ²)/5 - 5² = 10, so Σ xᵢ² = 175. Since 1 + 9 + 49 = 59, we get a² + b² = 116.
3. Find a and b. (a+b)² = 196, so 2ab = 196 - 116 = 80 and ab = 40. The numbers with sum 14 and product 40 are 10 and 4. Since a > b, a = 10 and b = 4.
4. New observations. n + xₙ gives 1+1, 2+3, 3+10, 4+7, 5+4, that is 2, 5, 13, 11, 9. Their sum is 40, so the new mean is 8 and the deviations are -6, -3, 5, 3, 1.

   σ² = (36 + 9 + 25 + 9 + 1)/5 = 80/5 = 16

5. So. The variance of the observations n + xₙ is 16, option (b).

The idea to keep. When each value gets a different number added, the variance does not stay the same. Find the actual data, here using a > b to fix the order, then compute the variance directly.

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