The mean and variance of 10 observations are 10 and 12 respectively. Later, it was found that one observation a was incorrect and it was replaced by the correct observation b . If the new mean and new variance are 10.5 and 10.25 respectively, then the value of a^2 + b^2 is equal to

Step-by-step solution

Let the initial sum of observations be x and the initial sum of squares be x^2 . Given N = 10 , initial mean = 10 , and initial variance = 12 . x 10 = 10 x = 100 x^2 10 - (10)^2 = 12 x^2 = 1120 Let the replaced observation be a and the new observation be b . The new sum and new sum of squares are: x_ new = 100 - a + b x_ new ^2 = 1120 - a^2 + b^2 Given the new mean is 10.5 and the new variance is 10.25 : 100 - a + b 10 = 10.5 b - a = 5 1120 - a^2 + b^2 10 - (10.5)^2 = 10.25 1120 - a^2 + b^2 10 - 110.25 = 10.25 1120