Consider a data consisting of 10 observations x_1, x_2, , x_ 10 , whose mean is 5 and variance is 7. If the mean and the variance of the first 8 observations x_1, x_2, , x_8 are 4 and 3.5, respectively, and x_9 < x_ 10 , then the value of 3x_9 + 2x_ 10 is ___________.
A. A
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: x_9 + x_ 10 = 50 - 32 = 18 x_9^2 + x_ 10 ^2 = 320 - 156 = 164 Using (x_9 + x_ 10 )^2 = x_9^2 + x_ 10 ^2 + 2x_9 x_ 10 : 18^2 = 164 + 2x_9 x_ 10 2x_9 x_ 10 = 324 - 164 = 160 x_9 x_ 10 = 80 The numbers x_9 and x_ 10 are roots of the equation t^2 - 18t + 80 = 0. (t - 8)(t - 10) = 0 t = 8, 10 Since x_9 Therefore, 3x_9 + 2x_ 10 = 3(8) + 2(10) = 24 + 20 = 44. Answer: 44
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