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Read the following scenario and answer the THREE questions that follow.
The upper hinge of a dataset is the median of all the values to the right of the median of the dataset in an ascending arrangement, while the lower hinge of the dataset is the median of all the values to the left of the median of the dataset in the same arrangement.
For example, consider the dataset 4, 3, 2, 6, 4, 2, 7. When arranged in the ascending order, it becomes 2, 2, 3, 4, 4, 6, 7. The median is 4 (the bold value), and hence the upper hinge is the median of 4, 6, 7, i.e., 6. Similarly, the lower hinge is 2.
A student has surveyed thirteen of her teachers, and recorded their work experience (in integer years). Two of the values recorded by the student got smudged, and she cannot recall those values. All she remembers is that those two values were unequal, so let us write them as A and B, where A < B. The remaining eleven values, as recorded, are: 5, 6, 7, 8, 12, 16, 19, 21, 21, 27, 29. Moreover, the student also remembers the following summary measures, calculated based on all the thirteen values:
Minimum: 2
Lower Hinge: 6.5
Median: 12
Upper Hinge: 21
Maximum: 29
Based on the information recorded, which of the following can be the average work experience of the thirteen teachers?
The first clue is the minimum value provided:
That is not one of the eleven values that we already have; this must mean that one of the missing values, A or B, must be 2
Since we are given that A is less than B, it must be A, which is equal to 2.
Now, writing down the twelve known numbers,
2, 5, 6, 7, 8, 12, 16, 19, 21, 21, 27, 29
we are also provided that the median of the data set was 12, which would be possible if the value 12 was shifted from the 6th position to the 7th position.
This must mean that the other missing value would also be in the first 6 values of the data set, lying in the range $$2\le B\le12$$
Now, narrowing down our focus to the lower hinge, the value is given to be 6.5
The first six values would be 2, 5, 6, 7, 8 and one unknown value
The lower hinge would be 6.5 only if the missing value appears after 7
Narrowing down the possible values of B to be 7, 8, 9, 10, 11, 12
In order to find the average work-experience of all the teachers, we must add up all the values
adding up the known values we get
2+5+6+7+8+12+16+19+21+21+27+29 = 173
Looking at the options, the total sum of all 13 values must be
A: 156
B: 162.5
C: 169
D: 175.5
E: 182
The sum of 13 values can only be 182 of all the given options, which would be the case when B is 9
Therefore, option E is the correct answer.
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