The variance of 25 observations is 8 . If each observation is multiplied by 3 , then the new variance of the resulting observations is

Step-by-step solution

Let the observations be x₁, x₂, , x₂₅ . The variance of these observations is given by Var(X) = 1 n _ i=1 ^ n (x_i - x )^2 = 8 . When each observation is multiplied by a constant k , the new observations are y_i = kx_i . The new mean is y = k x . The new variance is Var(Y) = 1 n _ i=1 ^ n (y_i - y )^2 = 1 n _ i=1 ^ n (kx_i - k x )^2 . Factoring out k^2 , we get Var(Y) = k^2 ( 1 n _ i=1 ^ n (x_i - x )^2 ) = k^2 Var(X) . Given k = 3 and Var(X) = 8 , the new variance is 3^2 8 = 9 8 = 72 . Answer: 72