The mean and variance of 20 observations are 10 and 15 , respectively. It was later found that one observation was recorded incorrectly. When this incorrect observation is replaced by the correct observation, the new mean and variance become 10.2 and 14.96 , respectively. The product of the correct and incorrect observations is

Step-by-step solution

Let the incorrect observation be and the correct observation be . Given n = 20 , initial mean x = 10 , and initial variance ^2 = 15 . Initial sum of observations, x = 20 10 = 200 . Initial sum of squares, x^2 = 20( ^2 + x ^2) = 20(15 + 100) = 2300 . After replacement, new mean x _ new = 10.2 and new variance ^2_ new = 14.96 . New sum of observations = 20 10.2 = 204 . New sum of squares = 20(14.96 + (10.2)^2) = 20(14.96 + 104.04) = 20(119) = 2380 . From the sums of observations: - = 204 - 200 = 4 From the sums of sq