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
Which of the following is a possible value of B?
- 2
- 6
- 8
- 13
- 29
The Correct Option is C
Solution and Explanation
Based on the information recorded, which of the following can be the average work experience of the thirteen teachers?
- 12
- 12.5
- 13
- 13.5
- 14
The Correct Option is
Solution and Explanation
Step 1: Calculate the total sum of the dataset. The average is given by:
Average = $\frac{\text{Sum of all values}}{\text{Total number of values}}$.
Let A and B represent the smudged values, where A + B = 18 (from Question 20). The eleven known values are:
5, 6, 7, 8, 12, 16, 19, 21, 21, 27, 29.
The sum of the eleven known values:
5 + 6 + 7 + 8 + 12 + 16 + 19 + 21 + 21 + 27 + 29 = 171.
Step 2: Calculate the total sum with A and B. The total sum is:
Total Sum = 171 + A + B = 171 + 18 = 189.
Step 3: Calculate the average.
Average = $\frac{189}{13} = 13$.
Final Answer: 13.
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?
What is the value of B?
- 7
- 9
- 10
- 12
- Cannot be determined from the given information.
The Correct Option is C
Solution and Explanation
Step 1: Determine the total recalculated sum. The recalculated average is 15. Using the formula for average:
Average = $\frac{\text{Sum of all values}}{\text{Total number of values}}$
15 = $\frac{\text{Recalculated Total Sum}}{13}$
Thus:
Recalculated Total Sum = 15 × 13 = 195.
Step 2: Calculate the correction applied. The original sum of the eleven known values was 171. After adding A and B, the total becomes:
171 + A + B = 171 + 18 = 189.
The recalculated total sum is 195, so the difference due to the correction is:
195 − 189 = 6.
This means one of the recorded values was half of its correct value. Let the wrongly recorded value be x. Then:
$\frac{x}{2}$ + 6 = x = => x = 12.
Step 3: Solve for B. From Question 20, A + B = 18. If A = 6, then:
B = 18 − 6 = 9.
Final Answer: 9.
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