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If the average (arithmetic mean) of dataset \(S\) is not greater than the average (arithmetic mean) of any non-empty subset of \(S\), which of the following statements must be true?
I. \(S\) contains only one element.
II. All elements in \(S\) are equal.
III. The median of \(S\) is equal to the average (arithmetic mean) of \(S\).
A. none of the three qualities is necessary
B. II only
C. III only
D. II and III only
E. I, II, and III
I. \(S\) contains only one element.
II. All elements in \(S\) are equal.
III. The median of \(S\) is equal to the average (arithmetic mean) of \(S\).
A. none of the three qualities is necessary
B. II only
C. III only
D. II and III only
E. I, II, and III
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04 May 2010, 14:00
Kudos
Bookmarks
Official Solution:
If the average (arithmetic mean) of dataset \(S\) is not greater than the average (arithmetic mean) of any non-empty subset of \(S\), which of the following statements must be true?
I. \(S\) contains only one element.
II. All elements in \(S\) are equal.
III. The median of \(S\) is equal to the average (arithmetic mean) of \(S\).
A. none of the three qualities is necessary
B. II only
C. III only
D. II and III only
E. I, II, and III
Consider a dataset \(S\). If the average of \(S\) is not greater than the average of ANY subset of \(S\), then the following scenarios are possible:
A. \(S=\{x\}\), where \(S\) contains only one element (e.g. {7});
B. \(S=\{x, x, ...\}\), where \(S\) contains more than one element and all elements are equal (e.g. {7,7,7,7}).
This is because if the dataset \(S\) contains two or more different elements, then we can always consider the subset with the smallest number, and the mean of this subset (mean of the subset = smallest number) will be less than the mean of the entire dataset (mean of the full dataset > smallest number).
For example, if \(S=\{3, 5\}\), then the mean of \(S=4\). If we pick the subset with the smallest number, \(s'=\{3\}\), then the mean of \(s'=3\). Thus, \(3 < 4\).
Now let's consider the statements:
I. \(S\) contains only one element. This statement is not always true since scenario B is also possible (\(S=\{x, x, ...\}\)).
II. All elements in \(S\) are equal. This statement is true for both scenarios A and B, hence always true.
III. The median of \(S\) is equal to the average (arithmetic mean) of \(S\). This statement is true for both scenarios A and B, hence always true.
Therefore, statements II and III are always true.
Answer: D
If the average (arithmetic mean) of dataset \(S\) is not greater than the average (arithmetic mean) of any non-empty subset of \(S\), which of the following statements must be true?
I. \(S\) contains only one element.
II. All elements in \(S\) are equal.
III. The median of \(S\) is equal to the average (arithmetic mean) of \(S\).
A. none of the three qualities is necessary
B. II only
C. III only
D. II and III only
E. I, II, and III
Consider a dataset \(S\). If the average of \(S\) is not greater than the average of ANY subset of \(S\), then the following scenarios are possible:
A. \(S=\{x\}\), where \(S\) contains only one element (e.g. {7});
B. \(S=\{x, x, ...\}\), where \(S\) contains more than one element and all elements are equal (e.g. {7,7,7,7}).
This is because if the dataset \(S\) contains two or more different elements, then we can always consider the subset with the smallest number, and the mean of this subset (mean of the subset = smallest number) will be less than the mean of the entire dataset (mean of the full dataset > smallest number).
For example, if \(S=\{3, 5\}\), then the mean of \(S=4\). If we pick the subset with the smallest number, \(s'=\{3\}\), then the mean of \(s'=3\). Thus, \(3 < 4\).
Now let's consider the statements:
I. \(S\) contains only one element. This statement is not always true since scenario B is also possible (\(S=\{x, x, ...\}\)).
II. All elements in \(S\) are equal. This statement is true for both scenarios A and B, hence always true.
III. The median of \(S\) is equal to the average (arithmetic mean) of \(S\). This statement is true for both scenarios A and B, hence always true.
Therefore, statements II and III are always true.
Answer: D
General Discussion
05 May 2010, 02:10
Kudos
Bookmarks
Bunuel
angel2009
If the mean of set S does not exceed mean of any subset of set S , which of the following must be true about set S ?
I. Set S contains only one element
II. All elements in set S are equal
III. The median of set S equals the mean of set S
A. none of the three qualities is necessary
B. II only
C. III only
D. II and III only
E. I, II, and III
I. Set S contains only one element
II. All elements in set S are equal
III. The median of set S equals the mean of set S
A. none of the three qualities is necessary
B. II only
C. III only
D. II and III only
E. I, II, and III
"The mean of set S does not exceed mean of any subset of set S" --> set S can be:
A. \(S=\{x\}\) - S contains only one element (eg {7});
B. \(S=\{x, x, ...\}\) - S contains more than one element and all elements are equal (eg{7,7,7,7}).
Why is that? Because if set S contains two (or more) different elements, then we can always consider the subset with smallest number and the mean of this subset (mean of subset=smallest number) will be less than mean of entire set (mean of full set>smallest number).
Example: S={3, 5} --> mean of S=4. Pick subset with smallest number s'={3} --> mean of s'=3 --> 3<4.
Now let's consider the statements:
I. Set S contains only one element - not always true, we can have scenario B too (\(S=\{x, x, ...\}\));
II. All elements in set S are equal - true for both A and B scenarios, hence always true;
III. The median of set S equals the mean of set S - - true for both A and B scenarios, hence always true.
So statements II and III are always true.
Answer: D.
Why can't we consider the subset with the greatest number
Ex- S= (5,6,7) and subset s (7)
Please explain
