Let in a series of 2 n observations, half of them are equal to a and remaining half are equal to - a . Also by adding a constant b in each of these observations, the mean and standard deviation of new set become 5 and 20 , respectively. Then the value of a 2 + b 2 is equal to :

Step-by-step solution

Let observations are denoted by x i for 1 ≤ i < 2 n x ¯ = ∑ x i 2 n = ( a + a + … + a ) - ( a + a + … + a ) 2 n ⇒ x ¯ = 0 and σ x 2 = ∑ x i 2 2 n - x ¯ 2 = a 2 + a 2 + … + a 2 2 n - 0 = a 2 ⇒ σ x = a Now, adding a constant b then y ¯ = x ¯ + b = 5 ⇒ b = 5 and σ y = σ x (No change in S.D.) ⇒ a = 20 ⇒ a 2 + b 2 = 425