For a set of 20 observations x₁, x₂, , x₂₀ , the sum of the squares of their deviations from a constant c is minimum when c = 10 , and this minimum value is 320 . Later, it was discovered that one observation was incorrectly recorded as 15 instead of -5 . Let x'₁, x'₂, , x'₂₀ be the corrected observations. A new series is defined as y_i = 3x'_i - 2 for i = 1, 2, , 20 . If and ^2 are the mean and the variance of y_i r

Step-by-step solution

The sum of squared deviations (x_i - c)^2 is minimized when c is equal to the mean of the observations. Thus, the old mean is _x = 10 . The minimum value of the sum of squared deviations is n Var (x) . So, 20 Var (x) = 320 Var (x) = 16 . The old sum of observations is x_i = 20 10 = 200 . The corrected sum of observations is x'_i = 200 - 15 + (-5) = 180 . The corrected mean is _ x' = 180 20 = 9 . The old sum of squares is x_i^2 = 20( Var (x) + _x^2) = 20(16 + 100) = 2320 . The corrected sum of squares is (x'_i)^2 =