The mean and variance of 8 observations 4, 4, 6, 8, 12, 12, a, b (where a < b ) are 7 and 12 respectively. A new set of 8 observations is defined as y_i = x_i^2 - ax_i + b , where x_i are the original observations. The mean of the new set of observations y₁, y₂, , y₈ is :

Step-by-step solution

Given the mean of the observations is 7 : 4 + 4 + 6 + 8 + 12 + 12 + a + b 8 = 7 46 + a + b = 56 a + b = 10 Given the variance is 12 : 4^2 + 4^2 + 6^2 + 8^2 + 12^2 + 12^2 + a^2 + b^2 8 - (7)^2 = 12 16 + 16 + 36 + 64 + 144 + 144 + a^2 + b^2 8 = 61 420 + a^2 + b^2 = 488 a^2 + b^2 = 68 Using (a+b)^2 = a^2 + b^2 + 2ab : (10)^2 = 68 + 2ab 100 = 68 + 2ab ab = 16 Solving a+b=10 and ab=16 with a The new observations are given by the transformation y_i = x_i^2 - 2x_i + 8 . The mean of y_i is the expected value E[Y] : y = 1 8