Let the mean and the variance of 6 observations a, b, 68,44,48,60 be 55 and 194, respectively. If a>b, then a+3 b is

Solution

Maths sir · 4 steps

Work with deviations from the mean 55: then a and b sit symmetrically on either side of it.

1. Use the mean. The total is 6 × 55 = 330. The four known values add to 68 + 44 + 48 + 60 = 220, so a + b = 110. That means (a - 55) + (b - 55) = 0, so the two deviations are equal and opposite.

   Σ (xᵢ - 55)² = 6 × 194 = 1164

2. Known deviations. The four known values deviate by 13, -11, -7, 5. Their squares add to 169 + 121 + 49 + 25 = 364. So (a - 55)² + (b - 55)² = 1164 - 364 = 800.
3. Find a and b. Put a - 55 = d and b - 55 = -d. Then 2d² = 800, so d = ± 20. Since a > b, d = 20, which gives a = 75 and b = 35.
4. So. a + 3b = 75 + 3 × 35 = 75 + 105 = 180, option (a).

The idea to keep. When two values are missing, measure everything from the mean. The deviations add to zero and their squares add to nσ², which gives you both equations at once.

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