If x is the mode of a set with 5 elements and x is less than exactly

Question

If x is the mode of a set with 5 elements and x is less than exactly two members of the set, which of the following must be true?
 
 I. x is the median of the set

II. The median of the set is greater than the mean of the set

III. x is less than the greatest member of the set

A. None
B. I only
C. III only
D. I and III only
E. I, II and III

Kurtosis · Answer

If x is the mode of a set with 5 elements and x is less than exactly two members of the set --> 
Possible sets in ascending order of values - {y, x, x, two elements} or {x, x, x, two elements} --> Clearly x must be the median and x must be less than the largest value of the set.

The median of the above two sets can be greater or lesser than the mean of the set. Eg: {1, 1, 1, 2, 2} or {1, 99, 99, 100, 101}

Answer: D

JeffTargetTestPrep · Answer

Since x is the mode of a set with 5 elements, there must be at least 2 elements in the set that are x. Since x is less than exactly two members of the set, there must be either 2 elements or 3 elements in the set that are x. Therefore, the possible scenarios for the 5 elements (from smallest to largest) are (note: we are assuming each variable below has a unique value):

1) w, x, x, y, z
2) x, x, x, y, z
3) x, x, x, y, y

Now, let’s analyze each Roman numeral.

I. x is the median of the set.

We can see that in all 3 possible scenarios, x is the median of the set. Roman numeral I is true.

II. The median of the set is greater than the mean of the set.

Since we don’t know actual values of the 5 elements in the set, Roman numeral II might be or might not true. For instance, if we use scenario 3 above and x = 5 and y = 10, the median = 5, while the mean is (5+5+5+10+10)/5 = 35/5 = 7.

III. x is less than the greatest member of the set.

Looking at our 3 scenarios, we see that x is always less than the greatest member of the set. Roman numeral III is true.

Answer: D

KarishmaB · Answer

A set with 5 elements has a mode - it means that at least one element is repeated. 
The mode is less than exactly two members. So in increasing order it could look something like this:

____, x, x, 10, 15

Where x is the mode and less than 10. The first element could be x or less than x.

I. x is the median of the set
Yes, the third element is x so it will be the median.

II. The median of the set is greater than the mean of the set
The median of the set is x. Mean could be anything since we have no limitations on the least and the greatest elements. Not necessary.

III. x is less than the greatest member of the set
x is less than 10 so obviously, it is less than the greatest element of the set too.

Answer (D)

bumpbot · Answer

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If x is the mode of a set with 5 elements and x is less than exactly

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