Let the mean and the variance of 5 observations x 1 , x 2 , x 3 , x 4 , x 5 be 24 5 and 194 25 respectively. If the mean and variance of the first 4 observation are 7 2 and a respectively, then 4 a + x 5 is equal to

Step-by-step solution

Mean x ¯ = ∑ x i 5 = 24 5 ⇒ ∑ x i = 24       . . . i Variance σ 2 = ∑ x i 2 5 - 24 5 2 = 194 25 ⇒ ∑ x i 2 = 154       . . . i i Also given mean x 1 + x 2 + x 3 + x 4 4 = 7 2 ⇒ x 1 + x 2 + x 3 + x 4 = 14 ⇒ x 5 = 10 (from i ) and variance σ 2 = x 1 2 + x 2 2 + x 3 2 + x 4 2 4 - 49 4 = a ⇒ x 1 2 + x 2 2 + x 3 2 + x 4 2 = 4 a + 49 ⇒ x 5 2 = 154 - 4 a - 49 (from i i ) ⇒ 100 = 105 - 4 a ⇒ 4 a = 5 Hence 4 a