Data set P consists of 30 values: six 10s, six 14s, six 18s, six 22s, and six 26s. Data set...

1. TRANSLATE the problem information

• Given information:
  • Data set P: six 10s, six 14s, six 18s, six 22s, six 26s (30 values total)
  • Data set Q: same as P, but four 10s become 26s and two 14s become 22s
• Need to compare the means of P and Q

2. INFER the most efficient approach

• We could calculate both means directly, but there's a smarter way
• Since we know exactly which values change, we can calculate how much the sum increases
• The mean will increase by: (increase in sum) ÷ (number of values)

3. SIMPLIFY to find the mean of P

• Calculate sum of P: \(6(10) + 6(14) + 6(18) + 6(22) + 6(26) = 540\)
• Mean of P = \(540 \div 30 = 18\)

4. SIMPLIFY to find the change in sum

• Four 10s become 26s: each gains \(26 - 10 = 16\), so total gain = \(4 \times 16 = 64\)
• Two 14s become 22s: each gains \(22 - 14 = 8\), so total gain = \(2 \times 8 = 16\)
• Total sum increase = \(64 + 16 = 80\)

5. SIMPLIFY to find the mean of Q

• Sum of Q = \(540 + 80 = 620\)
• Mean of Q = \(620 \div 30 = 20.67\)

Since \(20.67 \gt 18\), the mean of Q is greater than the mean of P.

Answer: (A)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misread the transformations, thinking that "changing four of the 10s to 26s" means adding four more 26s instead of replacing four 10s. This leads to incorrectly thinking Q has 34 values instead of 30.

When they try to calculate with wrong totals, they get confused about the denominator and either get stuck or make calculation errors that lead to guessing.

Second Most Common Error:

Poor INFER reasoning: Students calculate both complete datasets instead of using the sum-change method. While this approach works, it's more prone to arithmetic errors because it involves calculating:

• Q: two 10s, four 14s, six 18s, eight 22s, ten 26s

The multiple calculations (\(2 \times 10 + 4 \times 14 + 6 \times 18 + 8 \times 22 + 10 \times 26\)) create more opportunities for mistakes, potentially leading them to select Choice (B) or (C) based on calculation errors.

The Bottom Line:

This problem rewards students who recognize that data transformations can be analyzed by tracking changes rather than recalculating everything. The key insight is that moving values from smaller numbers to larger numbers will always increase the mean.