Question

Melanie computes the mean $\mu,$ the median $M,$ and the modes of the $365$ values that are the dates in the months of $2019.$ Thus her data consist of $12$ copies of $1,$ $12$ copies of $2,$ . . . , $12$ copies of $28,$ $11$ copies of $29,$ $11$ copies of $30,$ and $7$ copies of $31.$ Let $d$ be the median of the modes. Which of the following statements is true?

$\mu \lt d \lt M$

$M \lt d \lt \mu$

$d = M =\mu$

$d \lt M \lt \mu$

$d \lt \mu \lt M$

Answer: E

Concepts:mean median (data)mode

Difficulty rating: 1370

Solution:

The modes are all the integers from $1$ through $28,$ so their median is $d=\dfrac{14+15}{2}=14.5.$

There are $365$ entries, so $M$ is the $183$ rd number. The dates from $1$ through $15$ occupy $15 \cdot 12 = 180$ positions, so $M=16.$

The sum of all dates is $\begin{aligned}12(1+\cdots+28)&+11(29+30)\\&+7\cdot31=5738.\end{aligned}$ Hence $\mu=\dfrac{5738}{365}\approx15.72.$

Therefore $d \lt \mu \lt M.$ Thus, E is the correct answer.

2019 AMC 10A Problem 12

Answer: E

Difficulty rating: 1370

Solution:

Problem 12 in Other Years