Re M17-33
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16 Sep 2014, 01:02
Official Solution:
List of numbers S consists of positive numbers. If "-1" is added to S as a new element, each of the following could be true EXCEPT:
A. The range of the numbers in the new list will increase.
B. The average (arithmetic mean) of the numbers in the new list will NOT change.
C. The mode of the numbers in the new list will NOT change.
D. The median of the numbers in the new list will change.
E. The standard deviation of the numbers in the new list will change.
Let's examine each option:
A. The range of the numbers in the new list will change.
The range of a list is the difference between its largest and smallest elements. Since S consisted only of positive numbers, adding a new element of -1 to it means that the smallest element changes from some positive number to -1. Therefore, the range will definitely increase.
B. The average (arithmetic mean) of the numbers in the new list will NOT change.
The average of a list is \(\frac{sum}{\text{number of terms}}\). Adding -1 to the list will decrease the sum by 1. Therefore, the new average will be \(\frac{\text{a smaller sum}}{\text{more terms}}\), which will be less than \(\frac{sum}{\text{number of terms}}\). Hence the average cannot remain the same; it must decrease.
C. The mode of the numbers in the new list will NOT change.
The mode is the number that occurs most frequently in a list. For example, the mode of the list {2, 3, 4, 4} is 4. A list can have more than one mode. For example, the list {2, 2, 3, 3, 5} has two modes: 2 and 3. If every number in the list occurs an equal number of times, then the list has no mode. For example, the list {1, 2, 3} has no mode.
If S had a mode, say the number 3 occurring most frequently in the list, then adding -1 to the list would not change the mode: 3 would still be occurring most frequently in the list. Therefore, this option could be true.
D. The median of the numbers in the new list will change.
This is also possible. For example, if S = {1, 2, 3}, its median is 2, while S' = {-1, 1, 2, 3} and its median is 1.5. However, it's worth mentioning that even though it's possible for the median to change, it's not necessary to change. For example, if S = {1, 1, 1}, its median is 1. In this case, S' = {-1, 1, 1, 1}, and its median is also 1.
E. The standard deviation of the numbers in the new list will change.
The standard deviation of a list shows how much variation there is from the mean, how widespread a given list is. Adding -1 to a list of positive numbers will make the list more spread-out. Therefore, the standard deviation of the new list will definitely increase.
Answer: B