All but 5 students in a certain calculus class took the final exam on the scheduled day. The mean score...

Understanding the Question

Let's understand what we're being asked. We have:

• A calculus class where all but 5 students took the exam on the scheduled day
• Mean score for those who took it on schedule = \(\mathrm{m}\)
• Later, the 5 students took the exam
• Mean score for the entire class (everyone) = \(\mathrm{n}\)

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 \(\mathrm{n} < \mathrm{m}\).

Analyzing Statement 1

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}

• Mode = 70 (appears 3 times)
• Mean \(\mathrm{m} = 815 \div 10 = 81.5\)

Add 5 students with scores {95, 95, 95, 95, 95}

• New mode = 95 (now appears 6 times, higher than 70 ✓)
• New mean \(\mathrm{n} = (815 + 475) \div 15 = 1290 \div 15 = 86\)
• Result: \(\mathrm{n} > \mathrm{m}\) ✓

Example 2 - Where \(\mathrm{n} < \mathrm{m}\):
Original 10 students have scores {70, 80, 80, 80, 85, 85, 90, 90, 95, 95}

• Mode = 80 (appears 3 times)
• Mean \(\mathrm{m} = 850 \div 10 = 85\)

Add 5 students with scores {90, 90, 90, 70, 70}

• New mode = 90 (now appears 5 times, higher than 80 ✓)
• New mean \(\mathrm{n} = (850 + 410) \div 15 = 1260 \div 15 = 84\)
• Result: \(\mathrm{n} < \mathrm{m}\) ✓

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.

Analyzing Statement 2

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 \(\mathrm{m} = 80\)

• All 5 additional students score 100
• Range remains \(100 - 60 = 40\) ✓
• These 5 students averaged 100, which is > 80
• Therefore, \(\mathrm{n} > \mathrm{m}\) ✓

Example 2 - Where \(\mathrm{n} < \mathrm{m}\):
Same original students (60 to 100, \(\mathrm{m} = 80\))

• All 5 additional students score 60
• Range remains \(100 - 60 = 40\) ✓
• These 5 students averaged 60, which is < 80
• Therefore, \(\mathrm{n} < \mathrm{m}\) ✓

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.

Combining Statements

Now let's use both statements together:

• The mode increased when the 5 students were added
• The range stayed the same

Can we still get different outcomes?

Example 1 - Where \(\mathrm{n} > \mathrm{m}\):
Original scores {60, 70, 70, 70, 80, 90, 100} (7 students)

• Mode = 70, Range = 40, Mean \(\mathrm{m} = 520 \div 7 \approx 74.3\)

Add 5 scores: {90, 90, 90, 90, 90}

• New mode = 90 (higher than 70 ✓)
• Range = \(100 - 60 = 40\) (same ✓)
• New mean \(\mathrm{n} = (520 + 450) \div 12 = 970 \div 12 \approx 80.8\)
• Result: \(\mathrm{n} > \mathrm{m}\) ✓

Example 2 - Where \(\mathrm{n} < \mathrm{m}\):
Original scores {60, 80, 80, 90, 90, 90, 100} (7 students)

• Mode = 90, Range = 40, Mean \(\mathrm{m} = 590 \div 7 \approx 84.3\)

Add 5 scores: {100, 100, 100, 100, 60}

• New mode = 100 (higher than 90 ✓)
• Range = \(100 - 60 = 40\) (same ✓)
• New mean \(\mathrm{n} = (590 + 460) \div 12 = 1050 \div 12 = 87.5\)
• Wait, that gives \(\mathrm{n} > \mathrm{m}\). Let's adjust...

Add 5 scores: {100, 100, 100, 60, 60}

• New mode = 100 (higher than 90 ✓)
• Range = \(100 - 60 = 40\) (same ✓)
• New mean \(\mathrm{n} = (590 + 380) \div 12 = 970 \div 12 \approx 80.8\)
• Result: \(\mathrm{n} < \mathrm{m}\) ✓

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!]

The Answer: E

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."