Consider the first 10 natural numbers. If we multiply each number by - 1 and add 1 to each number, the variance of the numbers so obtained is

Step-by-step solution

Let the original set of numbers be X = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 . The variance of the first n natural numbers is given by the formula Var(X) = n^2 - 1 12 . For n = 10 , Var(X) = 10^2 - 1 12 = 99 12 = 8.25 . The new set of numbers is obtained by the transformation Y = -X + 1 . The property of variance states that Var(aX + b) = a^2 Var(X) . Here, a = -1 and b = 1 . Therefore, Var(Y) = (-1)^2 Var(X) = 1 8.25 = 8.25 . Answer: 8.25