A bag containing 10,000 beads of assorted colors is purchased from a craft store. To estimate the percent of red...

1. TRANSLATE the problem information

• Given information:
  • Total beads: 10,000
  • Estimated percent red: 15%
  • Margin of error: 2%
• What this tells us: We need to find how many actual red beads could plausibly be in the bag

2. TRANSLATE the percentage estimate into actual beads

• Calculate estimated red beads: \(15\% \times 10{,}000\)
• \(0.15 \times 10{,}000 = 1{,}500\) red beads (estimated)

3. TRANSLATE the margin of error into actual beads

• Calculate margin of error: \(2\% \times 10{,}000\)
• \(0.02 \times 10{,}000 = 200\) beads

4. INFER how margin of error creates a range

• Margin of error means the actual value could be 200 beads above or below our estimate
• Lower bound: \(1{,}500 - 200 = 1{,}300\) beads
• Upper bound: \(1{,}500 + 200 = 1{,}700\) beads
• Plausible range: \(1{,}300 \lt \mathrm{r} \lt 1{,}700\)

5. APPLY CONSTRAINTS to select the correct answer choice

• Check each option against our calculated range:

1. \(\mathrm{r} \gt 1{,}700\) (exceeds our upper bound)
2. \(1{,}300 \lt \mathrm{r} \lt 1{,}700\) (matches exactly)
3. \(200 \lt \mathrm{r} \lt 1{,}500\) (incorrectly uses margin of error as lower bound)
4. \(\mathrm{r} \lt 1{,}300\) (below our lower bound)

Answer: B. \(1{,}300 \lt \mathrm{r} \lt 1{,}700\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Misunderstanding what margin of error represents in the context of the problem

Students might think the margin of error is an absolute lower bound rather than a deviation from the estimate. They see 'margin of error of 2%' and think this means there could be as few as 200 beads total, leading them to select Choice C (\(200 \lt \mathrm{r} \lt 1{,}500\)).

Second Most Common Error:

Poor TRANSLATE execution: Calculating percentages incorrectly or mixing up which percentage applies to what

Students might incorrectly calculate 15% of 10,000 or confuse the margin of error calculation, leading to wrong bounds. This can cause confusion and random answer selection.

The Bottom Line:

This problem tests whether students understand that margin of error creates a symmetric range around an estimate, not an absolute boundary. The key insight is that 'estimate ± margin of error' gives you the plausible range for the true value.