Find the mean and standard deviation of S sigma Xi ie mus a

Find the mean and standard deviation of S = sigma X_i (i-e, mu_s and sigma_s) and mu = 1/n sigmaX_i (i.e, mu_X and sigma_x for the following values of X_i (assume all values of X_i are independent): If all X_1, X_n have mean 10, standard deviation 5, and n = 10. If X_1,,X_5 have mean -2, standard deviation 1, and X_6,...,X_10 have mean 2, standard deviation 1. If all X_1,...,X_n have mean 3, standard deviation 10, and n = 2 If X_1,,X_10 have mean 0, standard deviation 4, and X_11,,X_20 have mean 3, standard deviation 2.

Solution

(a) S_a = mean(10) * n (10) = 100 Mu_a = 10

Mean(a) = (S_a+Mu_a)/2 = 55 , Standard Deviation(a) = square root { ( [100-55]²+[10-55]²)/2 } = 45

(b) S_b = mean(-2) * n (5) + mean(2) * n(5) = 0 Mu_b = 0 / 10 = 0

Mean(b) = (Sb+Mub)/2 = 0 , Standard Deviation(b) = square root { ( [0-0]²+[0-0]²)/2 } = 0

(c) S_c = mean(3) * n (2) = 6 Mu_c= 6/2 = 3

Mean(c) = (S_c+Mu_c)/2 = 4.5 , Standard Deviation(c) = square root { ( [6-4.5]²+[3-4.5]²)/2 } = 1.5

(d) S_d = mean(0) * n (10) + mean(3) * n(10) = 30 Mu_d = 30 / 20 = 1.5

Mean(d) = (S_d+Mu_d)/2 = (30+1.5)/2 = 15.75 ,

Standard Deviation(d) = square root { ( [30-15.75]²+[1.5-15.75]²)/2 } = 14.25