Consider a data consisting of 10 observations x₁, x₂, , x₁₀ , whose mean is 5 and variance is 7 . If the mean and the variance of the first 8 observations x₁, x₂, , x₈ are 4 and 3.5 , respectively, and x₉ < x₁₀ , then the value of 3x₉ + 2x₁₀ is ___________.

Step-by-step solution

Given _ i=1 ¹⁰ x_i 10 = 5 _ i=1 ¹⁰ x_i = 50 _ i=1 ¹⁰ x_i^2 10 - 5^2 = 7 _ i=1 ¹⁰ x_i^2 = 320 For the first 8 observations: _ i=1 ⁸ x_i 8 = 4 _ i=1 ⁸ x_i = 32 _ i=1 ⁸ x_i^2 8 - 4^2 = 3.5 _ i=1 ⁸ x_i^2 = 8(3.5 + 16) = 156 Subtracting the sums, we get: x₉ + x₁₀ = 50 - 32 = 18 x₉^2 + x₁₀^2 = 320 - 156 = 164 Using (x₉ + x₁₀)^2 = x₉^2 + x₁₀^2 + 2x₉ x₁₀ : 18^2 = 164 + 2x₉ x₁₀ 2x₉ x₁₀ = 324 - 164 = 160 x₉ x₁₀ = 80 The numbers x₉ and x₁₀ are roots of the equation t^2 - 18t + 80 = 0 . (t - 8)(t - 10) = 0 t = 8, 10 Since x₉ T