A company fills boxes with approximately 23 pounds of oranges. To test the accuracy of the filling process, 344 boxes...

1. TRANSLATE the problem information

• Given information:
  • Sample estimate: 23.1 pounds
  • Margin of error: 0.19 pounds
  • Need to find: Range of plausible values
• The phrase "margin of error of 0.19 pounds" means we calculate \(\mathrm{estimate \pm 0.19}\)

2. INFER what margin of error represents

• Margin of error tells us how far the true population average could reasonably be from our sample estimate
• This creates a range of plausible values centered on our estimate
• Values within this range are considered reasonable; values outside are not

3. SIMPLIFY to find the confidence interval bounds

• Lower bound: \(\mathrm{23.1 - 0.19 = 22.91}\) pounds
• Upper bound: \(\mathrm{23.1 + 0.19 = 23.29}\) pounds
• Range of plausible values: 22.91 to 23.29 pounds

Answer: A (between 22.91 pounds and 23.29 pounds)

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about margin of error: Students may think margin of error represents values that are NOT plausible, leading them to believe plausible values are outside the calculated range rather than within it.

This backwards thinking may lead them to select Choice B (less than 22.91 or greater than 23.29).

Second Most Common Error:

Weak INFER skill: Students may focus on comparing the estimate (23.1) to some arbitrary value mentioned in the choices rather than using the margin of error to create a confidence interval.

This causes them to get stuck on whether 23.1 is greater or less than specific values, potentially leading to Choice C or D without proper consideration of the margin of error.

The Bottom Line:

The key insight is understanding that margin of error creates a "zone of plausible values" around the sample estimate - students who miss this fundamental concept about statistical estimation will struggle with any interpretation involving confidence intervals.