Problem:
Last academic year Yolanda and Zelda took different courses that did not necessarily administer the same number of quizzes during each of the two semesters. Yolanda's average on all the quizzes she took during the first semester was points higher than Zelda's average on all the quizzes she took during the first semester. Yolanda's average on all the quizzes she took during the second semester was points higher than her average for the first semester and was again points higher than Zelda's average on all the quizzes Zelda took during her second semester. Which one of the following statements cannot possibly be true?
Answer Choices:
A. Yolanda's quiz average for the academic year was points higher than Zelda's.
B. Zelda's quiz average for the academic year was higher than Yolanda's.
C. Yolanda's quiz average for the academic year was points higher than Zelda's.
D. Zelda's quiz average for the academic year equaled Yolanda's.
E. If Zelda had scored points higher on each quiz she took, then she would have had the same average for the academic year as Yolanda.
Solution:
To see that choice (C) might be true, observe that Yolanda's average on quizzes for the whole academic year would have been 3 points higher than Zelda's if the number of quizzes in each course was the same each semester. And if all courses have the same number of quizzes, the same argument shows that choice ( ) is also possible.
While counterintuitive, choice (B) could be true; this is an example of Simpson's Paradox. Indeed, suppose Yolanda's average was during her first semester. Then Yolanda's average was her second semester and Zelda's averages were and her two semesters. Suppose Yolanda took 8 quizzes her first semester and 2 quizzes her second semester, while Zelda took 2 quizzes her first semester and 8 quizzes her second semester. Then Yolanda's average for the whole academic year would have been
and Zelda's would have been
so Zelda would have had the better average for the whole academic year. Choice (D) could be true in much the same way. Again suppose Yolanda took 8 quizzes her first semester and 2 quizzes her second semester, and suppose Zelda took quizzes her first semester and quizzes her second semester. Then their averages would both have been if and only if
This equation is equivalent to , which will be true if and (or and ).
This leaves choice (A) as the only possible answer to this problem. To see that Yolanda's average on quizzes for the whole academic year cannot have been 22 points higher than Zelda's, use the notation as above and observe that Yolanda's average for the whole academic year must have been between and and Zelda's must have been between and . The greatest possible difference in their averages for the academic year is therefore less than points and cannot be 22 points.
The problems on this page are the property of the MAA's American Mathematics Competitions