For a dataset of 16 observations x₁, x₂, , x₁₆ , it is given that _ i=1 ¹⁶ (x_i - 5) = 32 and _ i=1 ¹⁶ (x_i - 5)^2 = 256 . It was later discovered that two observations were recorded incorrectly. The sum and product of these two incorrect observations are 24 and 108 , respectively. If they are replaced by two correct observations whose sum and product are 24 and 140 , respectively, then the variance of the corrected

Step-by-step solution

Let y_i = x_i - 5 . The variance is independent of the change of origin. Initial variance of the data is given by: ^2 = y_i^2 16 - ( y_i 16 )^2 ^2 = 256 16 - ( 32 16 )^2 = 16 - 2^2 = 12 Let the incorrect observations be a and b , and the correct observations be c and d . Given a+b = 24 and ab = 108 . Given c+d = 24 and cd = 140 . Since the sum of the incorrect observations equals the sum of the correct observations ( 24 ), the overall sum of the observations remains unchanged. Therefore, the mean of the data does n