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Z is a set of positive numbers. The median of Z is greater than the me

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Bunuel
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Z is a set of positive numbers. The median of Z is greater than the me [#permalink] New post   11 Apr 2021, 20:45
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Z is a set of positive numbers. The median of Z is greater than the mean of Z. Which of the following must be true?

I. At least 50% of the numbers in Z are smaller than the median.
II. Less than 50% of the numbers in Z are greater than the median.
III. The median of Z is greater than the average of the largest and smallest numbers in Z.

A. II only
B. III only
C. II and III
D. I, II and III
E. None of the above
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Bunuel
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Re: Z is a set of positive numbers. The median of Z is greater than the me [#permalink] New post   18 Apr 2021, 20:15
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Bunuel wrote:
Z is a set of positive numbers. The median of Z is greater than the mean of Z. Which of the following must be true?

I. At least 50% of the numbers in Z are smaller than the median.
II. Less than 50% of the numbers in Z are greater than the median.
III. The median of Z is greater than the average of the largest and smallest numbers in Z.

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


I. At least 50% of the numbers in Z are smaller than the median. Not always true. Consider {1, 2, 2} --> (median=2) > (mean=5/3) --> only one number (1, so 33%) is less than the median. Discard.

II. Less than 50% of the numbers in Z are greater than the median. Not always true. Consider {1, 2, 3, 3} --> (median=2.5) > (mean=9/4) --> 50% of the numbers (3 and 3) are greater than the median. Discard.

III. The median of Z is greater than the average of the largest and smallest numbers in Z. Not always true. Consider {1, 1, 3, 3, 5} --> (median=3) > (mean=13/5) --> (median=3) = (1+5)/2=3. Discard.

Answer: E.
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Re: Z is a set of positive numbers. The median of Z is greater than the me [#permalink] New post   11 Apr 2021, 23:34
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Bunuel wrote:
Z is a set of positive numbers. The median of Z is greater than the mean of Z. Which of the following must be true?

I. At least 50% of the numbers in Z are smaller than the median.
II. Less than 50% of the numbers in Z are greater than the median.
III. The median of Z is greater than the average of the largest and smallest numbers in Z.

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


Here, we do not have any major restrictions on the numbers we can pick, so in all probability none of the case should be a MUST.



I. At least 50% of the numbers in Z are smaller than the median.
Let the set Z be {3,4,4,4}.....Median = 4 and mean = 15/4=3.75.......Median>mean
Only one element out of 4 is less than the Median ......Not true

II. Less than 50% of the numbers in Z are greater than the median.
Let the set Z be {1,4,5,5}.....Median = 4.5 and mean = 15/4=3.75.......Median>mean
But 50% of the elements are greater than the median. ......Not true

III. The median of Z is greater than the average of the largest and smallest numbers in Z.
Let the set Z be {2,2,2,3,3,3,5}.....Median = 3 and mean = 20/7=2.8.......Median>mean
Average of first and the last number = (2+5)/2=3.5>3....Not true


E
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Z is a set of positive numbers. The median of Z is greater than the me [#permalink] New post   11 Apr 2021, 23:47
Z is a set of positive numbers. The median of Z is greater than the mean of Z. Which of the following must be true?

I. At least 50% of the numbers in Z are smaller than the median.
IF all numbers are same, so this will not hold good- Clearly not a "must be true statement"

II. Less than 50% of the numbers in Z are greater than the median.

Median is the middle term in the ascending/descending sequence of a set-
If there are odd number of elements, one less than half can be more or equal, not necessarily more.
If there are even number of elements, and middle two terms are different - 50% can be greater than the mean-Hence, not must be true

III. The median of Z is greater than the average of the largest and smallest numbers in Z.

Clearly, not must be true

Answer is E in my opinion
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Re: Z is a set of positive numbers. The median of Z is greater than the me [#permalink] New post   14 Jul 2023, 11:33
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Re: Z is a set of positive numbers. The median of Z is greater than the me [#permalink]
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