Consider a set of 3 n numbers having variance 4 . In this set, the mean of first 2 n numbers is 6 and the mean of the remaining n numbers is 3 . A new set is constructed by adding 1 into each of the first 2 n numbers, and subtracting 1 from each of the remaining n numbers. If the variance of the new set is k , then 9 k is equal to ______.

Step-by-step solution

Let, the number be a 1 ,   a 2 ,   a 3 , . . . ,   a 2 n ,   b 1 ,   b 2 ,   b 3 , . . . ,   b n . Given, the mean of the first 2 n numbers is 6 and that of the remaining n numbers is 3 , then by using the formula for combined mean, we get, the mean of the complete set as μ = 2 n × 6 + n × 3 2 n + n = 15 n 3 n = 5 . Also, the variance is defined as σ 2 = ∑ x i 2 n - μ 2 4 = ∑ a i 2 + ∑ b i 2 3 n - ( 5 ) 2 ⇒ ∑ a i 2 + ∑ b