Problem:
Melanie computes the mean , the median , and the modes of the values that are the dates in the months of . Thus her data consists of , , . . . , , , , and . Let be the median of the modes. Which of the following statements is true?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Each of the values through is a mode, so . There are data values less than or equal to , and there are values less than or equal to . Therefore more than half of the values are greater than or equal to and more than half of the values are less than or equal to , so . To see the relationship between and , note that if every month had days, then there would be of each value from to , and the mean would be ; because the actual data are missing some of the larger values, . To see the relationship between and , note that if every month had days, then there would be of each value from to , and the mean would be ; because the actual data consist of all of these values together with some larger values, . Therefore .
The problems on this page are the property of the MAA's American Mathematics Competitions