In a study, the data from a random sample of a population had a mean of 37, with an associated...

1. TRANSLATE the problem information

• Given information:
  • Sample mean = 37
  • Margin of error = 3
  • We need to find what we can conclude about the population mean

2. INFER the approach

• A margin of error tells us how far the true population mean might be from our sample mean
• We construct a confidence interval: sample mean ± margin of error
• This gives us the range where the population mean is likely to fall

3. SIMPLIFY to find the interval bounds

• Lower bound: \(\mathrm{37 - 3 = 34}\)
• Upper bound: \(\mathrm{37 + 3 = 40}\)
• So the population mean is between 34 and 40

Answer: C. It is between 34 and 40.

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about margin of error: Students may think the margin of error tells them something definitive about where the population mean IS, rather than understanding it creates a range of likely values.

This misconception might lead them to think the population mean must be exactly 37, or that it must be different from 37, causing them to select Choice A or Choice B based on incomplete reasoning.

Second Most Common Error:

Weak INFER skill: Students may not understand what the margin of error actually represents statistically. They might interpret "margin of error of 3" as meaning the population mean is definitely NOT between 34 and 40.

This backwards reasoning leads them to select Choice D (It is less than 34 or greater than 40).

The Bottom Line:

The key insight is that margin of error creates uncertainty around our sample estimate - it doesn't tell us the population mean is different from our sample mean, but rather gives us a range of plausible values that includes our sample mean.