Based on a random sample from a population, a researcher estimated that the mean value of a certain variable for...

1. TRANSLATE the problem information

• Given information:
  • Sample-based estimate of population mean = 20.5
  • Margin of error = 1
• What we need: The most appropriate conclusion about the actual population mean

2. INFER what margin of error means

• Margin of error creates a range of plausible values for the true population parameter
• It does NOT create absolute boundaries - values outside this range are still possible, just less likely
• The range is: \(\mathrm{20.5 ± 1 = [19.5, 21.5]}\)

3. INFER the correct interpretation

• We can say it's 'plausible' or 'likely' that the true population mean falls within \(\mathrm{[19.5, 21.5]}\)
• We cannot say it's 'impossible' for the true mean to be outside this range
• This range refers to the population mean, not to individual data values

4. Evaluate answer choices using statistical reasoning

• Choice A: Uses correct language ('plausible') and correct range ✓
• Choice B: Uses incorrect absolute language ('not possible')
• Choice C: Confuses population mean with individual values
• Choice D: Treats sample estimate as exact population value

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students misunderstand margin of error as creating absolute boundaries rather than indicating plausible ranges.

They think: 'If the margin of error is 1, then the true mean absolutely cannot be outside \(\mathrm{[19.5, 21.5]}\).' This leads them to interpret statistical uncertainty as mathematical certainty.

This may lead them to select Choice B ('It is not possible that the mean value...is less than 19.5 or greater than 21.5')

Second Most Common Error:

Conceptual confusion: Students confuse the population mean (what we're estimating) with individual data values in the population.

They think: 'The margin of error tells us something about all the individual values in the population.' This fundamental misunderstanding about what parameters vs. data values represent causes confusion.

This may lead them to select Choice C ('Every value of the variable in the population is between 19.5 and 21.5')

The Bottom Line:

This problem tests whether students understand that statistical inference deals with uncertainty and plausibility, not absolute certainty, and whether they can distinguish between population parameters and individual data points.