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All but 5 students in a certain calculus class took the final exam on the scheduled day. The mean score for these students was \(\mathrm{m}\). Later, the 5 students who had missed the scheduled exam took it and the mean score for the entire class was calculated. If \(\mathrm{n}\) represents the mean score of the entire class, is \(\mathrm{n > m}\)?
Let's understand what we're being asked. We have:
We need to determine: Is \(\mathrm{n} > \mathrm{m}\)?
This is asking whether adding the 5 late students' scores raised or lowered the class average.
Here's the key insight: For \(\mathrm{n}\) to be greater than \(\mathrm{m}\), the 5 students who took the exam later would need to have an average score greater than \(\mathrm{m}\). If their average equals \(\mathrm{m}\), then \(\mathrm{n} = \mathrm{m}\). If their average is less than \(\mathrm{m}\), then \(\mathrm{n} < \mathrm{m}\).
Statement 1 tells us: The mode score for the entire class was higher than the mode score for the students who took the exam on the scheduled day.
The mode is the most frequently occurring score. While this tells us something changed in the distribution, it doesn't directly tell us about the mean.
Let's test with specific examples:
Example 1 - Where \(\mathrm{n} > \mathrm{m}\):
Original 10 students have scores {70, 70, 70, 80, 80, 85, 85, 90, 90, 95}
Add 5 students with scores {95, 95, 95, 95, 95}
Example 2 - Where \(\mathrm{n} < \mathrm{m}\):
Original 10 students have scores {70, 80, 80, 80, 85, 85, 90, 90, 95, 95}
Add 5 students with scores {90, 90, 90, 70, 70}
Since we found examples where \(\mathrm{n} > \mathrm{m}\) AND examples where \(\mathrm{n} < \mathrm{m}\), Statement 1 is NOT sufficient.
[STOP - Not Sufficient!]
This eliminates choices A and D.
Now let's forget Statement 1 completely and analyze Statement 2 independently.
Statement 2 tells us: The range of scores for the entire class was the same as the range for students who took the exam on the scheduled day.
\(\mathrm{Range} = (\mathrm{highest\ score} - \mathrm{lowest\ score})\). This means the 5 additional students didn't score higher than the highest original score or lower than the lowest original score.
Example 1 - Where \(\mathrm{n} > \mathrm{m}\):
Original students score from 60 to 100, with mean \(\mathrm{m} = 80\)
Example 2 - Where \(\mathrm{n} < \mathrm{m}\):
Same original students (60 to 100, \(\mathrm{m} = 80\))
The 5 students could score anywhere within the existing range, leading to different outcomes for \(\mathrm{n}\) vs \(\mathrm{m}\). Statement 2 is NOT sufficient.
[STOP - Not Sufficient!]
This eliminates choice B.
Now let's use both statements together:
Can we still get different outcomes?
Example 1 - Where \(\mathrm{n} > \mathrm{m}\):
Original scores {60, 70, 70, 70, 80, 90, 100} (7 students)
Add 5 scores: {90, 90, 90, 90, 90}
Example 2 - Where \(\mathrm{n} < \mathrm{m}\):
Original scores {60, 80, 80, 90, 90, 90, 100} (7 students)
Add 5 scores: {100, 100, 100, 100, 60}
Add 5 scores: {100, 100, 100, 60, 60}
Even with both conditions satisfied, we can construct examples where \(\mathrm{n} > \mathrm{m}\) and others where \(\mathrm{n} < \mathrm{m}\). The statements together are NOT sufficient.
[STOP - Not Sufficient!]
Neither statement alone nor both statements together provide enough information to determine whether \(\mathrm{n} > \mathrm{m}\).
Answer Choice E: "The statements together are not sufficient."