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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
While rechecking her original notes to re-enter the smudged values of A and B in the records, the student found that one of the eleven recorded work experience values that did not get smudged was recorded wrongly as half of its correct value. After re-entering the values of A and B, and correcting the wrongly recorded value, she recalculated all the summary measures. The recalculated average value was 15. What is the value of B?
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 the context of the third question, we are given that the average of the 13 values was 15, meaning that the sum of the 13 values was 195
As we calculated previously, the sum with the value we had would be 173 +B
we also have to add the mistaken value once more to this number to get 195
Let's take the unknown number to be U, giving us the equation
173 + B+ U = 195
B+U = 22
Where B must be one of 7, 8, 9, 10, 11 or 12
And U must be one of 5, 6, 7, 8, 12, 16, 19, 21, 21, 27, 29
trying to substitute the values of B into the equation, the only value which yields valid values for both B and U would be 10
Therefore, Option C is the correct answer.
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