Problem Solving Question: Descriptive Statistics

This question combines elementrary concepts of number properties including prime numbers, and properties of odd and even numbers with statistics. Elementary statistics concepts including range, mean and median of a set of numbers are tested in this question. It is extremely important in such questions to note down information such as "what kind of numbers is the set talking about", "what values the variable can take", "are those numbers distinct" - because these information make a big difference in finding the correct answer.

Question

  • PS
  • Consider a set S = {2, 4, 6, 8, x, y} with distinct elements. If x and y are both prime numbers and 0 < x < 40 and 0 < y < 40, which of the following MUST be true? I. The maximum possible range of the set is greater than 33.
    II. The median can never be an even number.
    III. If y = 37, the average of the set will be greater than the median.

    A. I only
    B. I and II only
    C. I and III only
    D. III only
    E. I, II, and III

 

  • Correct Answer
    Choice E. All 3 statements are true.

Explanatory Answer

Detailed Solution

Key Data from the Question Stem

  1. Set S has 6 elements.
  2. The elements of set S are distinct.
  3. x and y are prime numbers. Because 2 is already an element in S, both x and y have to be odd.
  4. 0 < x < 40 and 0 < y < 40
I. The maximum possible range of the set is greater than 33

The key word in this entire statement is maximum. We have to determine whether the maximum value possible for the range exceeds 33.

We know x and y are prime numbers. The largest prime number less than 40 is 37.
If either x or y is 37, the largest number in the set will be 37 and the smallest number is 2.
Therefore, the maximum range of the set will be 37 - 2 = 35. It is greater than 33.

Statement I is true. So, eliminate choices that donot contain I. Eliminate choice D

II. The median can never be an even number

There are 6 numbers in the set. Therefore, the median is the arithmetic mean of the 3rd and the 4th term when the numbers are written in ascending or descending order.

The elements are {2, 4, 6, 8, x, y}, where x and y are prime numbers.

If x and y take 3 and 5 as values, the median is 4.5
If x = 3, y = 7 or greater, the median is 5
If x = 5, y = 7 or greater, the median is 5.5
If x = 7, y = 11 or greater, the median is 6.5
If x = 11 or greater and y = 13 or greater, the median is 7.

It is quite clear that the median is either an odd number or is not an interger. So, the median can never be an even integer.

Statement II is true. Eliminate choices that do not contain II. Eliminate choices A and C as well.

III. If y = 37, the average of the set will be greater than the median

If y = 37, the set will be {2, 4, 6, 8, x, 37}, where x is a prime number greater than 2 and less than 37.

The average will be \\frac {57 + x}{6} \\) = 9.5 + \\frac {x}{6} \\)

If x = 3, median = 5 and average = 10. Average > median.
If x = 5, median = 5.5 and average = 10.33. Average > median
If x = 7, median = 6.5 and average = 10.66. Average > medain
If x = 11 or greater, the median = 7. Average will be definitely greater than 10. So, Average > Median.

It is true that the average is greater than the median if y = 37.

Statement III is also true.

Statements I, II, and III are true. Choice E is the answer.