Given that U = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}
Set A = { 8 different integers from U}
Target question => The standard deviation of set A
Standard deviation is a measure of each integer from the mean so we can evaluate the SD after knowing the 8 integers in set A or the 2 integers that are not in set A. Since there are 10 integers to select the 8 integers from.
Statement 1 => Set A does not include 11
Out of the 10 numbers in the universal set provided, 11 is not included in the new set A, there is one other number that is not in the new set A but no information is provided about that so statement 1 is NOT SUFFICIENT
Statement 2 => The average of the numbers in the give list is equal to the average of numbers in set A
Average of universal set = average of set A
$$\frac{\left(1+3+5+7+9+11+13+15+17+19\right)}{10}=\frac{sum\ of\ 8\ integers}{8}$$
$$sum\ of\ 8\ integers\ =\ \frac{100\cdot8}{10}=80$$
difference between the sum of universal set and set A = 100 - 80 = 20
So there are 2 numbers whose sum = 20 and are not included in the set from the numbers in the universal set, possible number that fits the information are [ 7, 13], [9,11], [13, 7], [1, 19], or [5, 15].
Since the 2 numbers not included in set A is unknown, then we cannot identify all the 8 numbers in set A. So, the standard deviation cannot be evaluated, statement 2 is NOT SUFFICIENT
Combining both statements together =>
From statement 1 => 11 is not included in set A
From statement 2 => possible list of 2 numbers not included are [7, 13], [9, 11], [13, 7], [1, 19], or [5,15]
From these 2 statements, the 2 numbers not included are 9 and 11 so set A = {1, 3, 5, 7, 9, 13, 15, 17, 19}
Since we know all the numbers in set A, it is now possible to evaluate the standard deviation by calculating the mean, subtracting it from each number then take the square differences before finding the square root.
Both statements combined together IS SUFFICIENT
Answer = C
