Data set A consists of the heights of 75 objects and has a mean of 25 meters. Data set B...

1. TRANSLATE the problem information

• Given information:
  • Data set A has 75 objects with \(\mathrm{mean = 25\ meters}\)
  • Data set B has 50 objects with \(\mathrm{mean = 65\ meters}\)
  • Data set C combines all objects from A and B
• What we need: Mean of data set C

2. INFER the solution approach

• To find the mean of combined data set C, we need:
  • Total sum of all values in C
  • Total number of values in C (which is \(\mathrm{75 + 50 = 125}\))
• Key insight: We can find the sum for each individual data set using the relationship: \(\mathrm{sum = mean \times number\ of\ values}\)

3. SIMPLIFY to find the sum of data set A

• \(\mathrm{Sum\ of\ A = mean \times number\ of\ values}\)
• \(\mathrm{Sum\ of\ A = 25 \times 75 = 1{,}875\ meters}\)

4. SIMPLIFY to find the sum of data set B

• \(\mathrm{Sum\ of\ B = mean \times number\ of\ values}\)
• \(\mathrm{Sum\ of\ B = 65 \times 50 = 3{,}250\ meters}\)

5. INFER how to combine the data sets

• For data set C (the combination):
  • Total sum = Sum of A + Sum of B = \(\mathrm{1{,}875 + 3{,}250 = 5{,}125\ meters}\)
  • Total count = \(\mathrm{75 + 50 = 125\ objects}\)

6. SIMPLIFY to find the final answer

• \(\mathrm{Mean\ of\ C = Total\ sum \div Total\ count}\)
• \(\mathrm{Mean\ of\ C = 5{,}125 \div 125 = 41\ meters}\) (use calculator)

Answer: 41

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to find the mean by averaging the two given means: \(\mathrm{(25 + 65) \div 2 = 45}\)

They incorrectly assume that the mean of combined data sets equals the average of the individual means, forgetting that the data sets have different sizes (75 vs 50 objects). This weighted difference is crucial - data set B's higher mean (65) should influence the combined mean less than data set A's lower mean (25) because B has fewer objects.

This may lead them to select an answer choice of 45 if available, or causes confusion when 45 doesn't appear as an option.

Second Most Common Error:

Missing conceptual knowledge: Students don't remember or apply the relationship \(\mathrm{sum = mean \times count}\) to find individual data set totals.

Without this key relationship, they get stuck trying to work directly with the means and can't figure out how to properly combine the data sets. They may attempt incorrect approaches like weighted averages without the proper weights.

This leads to confusion and guessing among the available answer choices.

The Bottom Line:

This problem tests whether students understand that combining data sets requires working with totals (sums), not just averaging the means. The key insight is recognizing that different-sized data sets contribute differently to the combined mean - it's a weighted average where the weights are the number of objects in each set.