The mean and variance of seven observations are 8 and 16, respectively. If 5 of the observations are 2, 4, 10, 12, 14, then the product of the remaining two observations is:

Using the mean formula, the sum of all 7 observations is 7 multiplied by 8, which is 56. The sum of the 5 given observations is 2 plus 4 plus 10 plus 12 plus 14, equaling 42, so the sum of the remaining two observations is 56 minus 42, or 14. Let the remaining observations be x and y, giving x plus y equals 14. The variance formula is the sum of squared deviations divided by the number of observations, so 7 times 16 equals 112. The sum of the squares of all observations equals 112 plus 7 times 64, yielding 560. The sum of squares of the 5 known values is 4 plus 16 plus 100 plus 144 plus 196, which is 460, leaving the sum of squares of x and y as 100. Substitute y equals 14 minus x into the equation x squared plus y squared equals 100 to get x squared plus 196 minus 28x plus x squared equals 100. Solving 2x squared minus 28x plus 96 equals 0 yields x squared minus 14x plus 48 equals 0. The roots of this quadratic equation are 6 and 8, so the product of the two observations is 6 multiplied by 8, giving 48.

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