Part A
Answer: 3
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Work Shown:
Original Set = {2, 8, 2, 1, 0, 0, 3, 5, 1, 8, 6, 5, 5}
Sorted set = {0, 0, 1, 1, 2, 2, 3, 5, 5, 5, 6, 8, 8}
There are 13 items in the sorted set, so the middle value will have 13/2 = 6.5 --> 6 items below it and 6 items above it. This means the median is in slot 6+1 = 7. For the sorted set, the number in slot 7 is the value 3
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Part B
Answer: one pair is 4 and 5
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Work Shown:
The old median (3) increases to 5 since we're told the median increases by 2. This means that 5 must be in the middle slot. Why? Because we go from 13 items to 15 items (again adding 2 numbers on). The set
{0, 0, 1, 1, 2, 2, 3, 5, 5, 5, 6, 8, 8}
becomes
{0, 0, 1, 1, 2, 2, 3, X, Y, 5, 5, 5, 6, 8, 8}
where 3 <= X <= Y
and X is the value of the new median. We must make X = 5 if we want to have this particular configuration.
The value of Y must be 5 as well
We'd have this new set
{0, 0, 1, 1, 2, 2, 3, 5, 5, 5, 5, 5, 6, 8, 8}
and the median would be 5.
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Part C
If you had the set {1,2,3,4} then the values 2 and 3 are tied for the middle-most position. Find the midpoint of these two values
midpoint = (A+B)/2 = (2+3)/2 = 5/2 = 2.5
for this example, the median is 2.5 as its directly in the middle of 2 and 3
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Part D
looking back at the original sorted set
{0, 0, 1, 1, 2, 2, 3, 5, 5, 5, 6, 8, 8}
the min and max are 0 and 8 respectively, so the range is
range = max - min
range = 8-0
range = 8
If we add in values that are between 0 and 8, then this will not affect the range. We would need to either add values that are larger than 8, or smaller than 0. Considering the values represent the number of runs, this means a negative number of runs makes no sense. So its only possible to have runs exceed 8.
If we had this set
{0, 0, 1, 1, 2, 2, 3, 5, 5, 5, 6, 8, 8, 22}
Then the range is now max-min = 22-0 = 22
But if we had this set
{0, 0, 1, 1, 2, 2, 3, 5, 5, 5, 6, 8, 8, 8}
then the range is still 8
So again, all you need to do is make sure you add in values that are between 0 and 8.
