Set X consists of 9 total terms, but only two different terms. Six of the terms are each equal to twice the value of each of the remaining 3. Which of the following would provide sufficient additional information to determine the average of the set?
Indicate all such statements.
1) The smaller number is positive and is 3 less than the larger number.
2) The standard deviation of the set is equal to 2 root 3.
3) The biggest term in the set is 6.
As per the answer key, the second option is incorrect. (other 2 are correct)
However, If we assume the smaller terms to be x each => larger terms = 2x each. The mean can be found out in terms of x, and hence the standandard deviation can also be found out in terms of x. This would give an equation of type: (something) x = 2(root 3) => x can be determined => average can be determined.
Is the book's answer incorrect then?
Indeed, the average is $\frac{6x+3\cdot 2x}{9}=\frac{4}{3}x$ and the snatdard deviation is $\sqrt{\frac{6\cdot\frac{1}{9}x^2+3\cdot\frac{4}{9}x^2}{9}} = \frac{\sqrt{2}}{3}|x|$ so the sign is missing. I.e. $x=\pm 3\sqrt{6}$ and we can't determine the average.