Let X₁, X₂, ......., X₁₈ be eighteen observations such
that Σᵢ₌₁¹⁸ (Xᵢ - α ) = 36 and Σᵢ₌₁¹⁸ (Xᵢ - β )² = 90, where α and β are distinct real numbers. If the standard deviation of these observations is 1, then the value of | α - β | is ____________.

Solution

Maths sir · 5 steps

Both sums tell you how far the mean is from a shift point. Find the mean, then use variance about any point.

1. Mean from alpha. From Σ (Xᵢ - α) = 36 we get Σ Xᵢ - 18α = 36. Dividing by 18 gives X̄ - α = 2, so X̄ = α + 2.
2. Variance about beta. The variance is the mean square deviation about any point β, minus the square of the mean's distance from β.

   σ² = 1/nΣᵢ₌₁ⁿ(Xᵢ-β)² - (X̄-β)²

3. Put in numbers. 1 = 90/18 - (X̄-β)² = 5 - (X̄-β)², so (X̄-β)² = 4 and X̄ - β = ± 2.
4. Use distinctness. With X̄ = α + 2, the root +2 gives α = β, which is not allowed. The root -2 gives α + 2 - β = -2, so α - β = -4.
5. So. |α - β| = 4.

The common slip: 0. Taking the root X̄ - β = +2 makes α = β and gives 0. The question says α and β are distinct, so that root must be rejected.

The idea to keep. Variance does not change with a shift, so σ² = 1/nΣ (xᵢ - a)² - (x̄ - a)² works for any a. Use it whenever the sums are taken about some point other than the mean.

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