Let X={11,12,13,....,40,41} and Y={61,62,63,....,90,91} be the two sets of observations. If x̅ and y̅ are their respective means and σ² is the variance of all the observations in X∪ Y, then | x̅ + y̅ - σ ² | is equal to ____________.

Solution

Maths sir · 4 steps

Both sets are 31 consecutive integers, so each has variance 80. Then combine the two groups properly.

1. Means. X has 41 - 11 + 1 = 31 terms in an AP, so its mean is the middle value x̅ = (11 + 41)/2 = 26. In the same way Y has 31 terms and y̅ = (61 + 91)/2 = 76. So x̅ + y̅ = 102.
2. Each variance. For n consecutive integers the variance is (n² - 1)/12. With n = 31 this is 960/12 = 80, for X and for Y.

   σ² = (σₓ² + σ_y²)/2 + ((x̅ - m)² + (y̅ - m)²)/2

3. Combine. The two groups are the same size, so the overall mean is m = (26 + 76)/2 = 51. Each group mean is 25 away from it, so σ² = 80 + (625 + 625)/2 = 80 + 625 = 705.
4. So. | x̅ + y̅ - σ² | = |102 - 705| = 603.

The common slip: 22. Taking σ² as just the common within-group variance 80 ignores how far the group means sit from 51, and gives |102 - 80| = 22.

The idea to keep. The variance of combined data is the average spread inside each group plus the spread of the group means about the overall mean. Never drop that second part.

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