A set of 10 observations has an initial mean of 15 and a variance of 15. It is later discovered that one observation, a , was recorded incorrectly. When a is replaced by the correct observation b , the new mean becomes 16 and the new variance becomes 14. The product of the two observations, a b , is equal to

Step-by-step solution

Let the incorrect sum of observations be x_i and the incorrect sum of squares be x_i^2 . Given the incorrect mean is 15 and n = 10 : x_i 10 = 15 x_i = 150 Using the incorrect variance of 15: x_i^2 10 - (15)^2 = 15 x_i^2 10 - 225 = 15 x_i^2 10 = 240 x_i^2 = 2400 When a is replaced by b , the new mean is 16 and the new variance is 14. Let the correct sums be y_i and y_i^2 . y_i 10 = 16 y_i = 160 y_i^2 10 - (16)^2 = 14 y_i^2 10 - 256 = 14 y_i^2 10 = 270 y_i^2 = 2700 Relating the incorrect and correct sums: y_i = x_i -