A dataset X consisting of 10 observations has a calculated mean of 20 and a variance of 15 . A new dataset Y is formed using the relation y_i = ax_i + b (where a > 0 ). The calculated mean and variance of Y are 45 and 60 respectively. Later, it was discovered that one observation in X was incorrectly read as 12 instead of the correct value 22 . If and ^2 are the true correct mean and true correct variance of dataset

Step-by-step solution

First, we find the transformation parameters a and b using the incorrect statistics. Var ( Y ) = a^2 Var ( X ) 60 = a^2(15) a^2 = 4 . Since a > 0 , a = 2 . Mean ( Y ) = a Mean ( X ) + b 45 = 2(20) + b b = 5 . Next, we correct the statistics for dataset X . Incorrect sum of observations = 10 20 = 200 . Correct sum = 200 - 12 + 22 = 210 . Correct mean of X = 210 10 = 21 . Incorrect sum of squares of observations x_i^2 = 10 ( Variance + Mean ^2) = 10(15 + 400) = 4150 . Correct sum of squares = 4150 - (12)^2 + (22)^2 =