When a survey reports that 60% of voters support a candidate "with a margin of error of ±3%", it means the true percentage is likely between 57% and 63%. The Digital SAT tests your understanding of margin of error, confidence intervals, and how sample size affects precision.
Core Concepts
Margin of Error
The margin of error gives the range around a sample statistic within which the true population parameter likely falls.
Factors Affecting Margin of Error
- Sample size: larger sample → smaller margin of error (more precise).
- Confidence level: higher confidence → larger margin of error (wider interval).
- Population variability: more variability → larger margin of error.
Interpreting Confidence Intervals
"We are 95% confident that the true proportion is between 57% and 63%."
This means: if we repeated the survey many times, about 95% of the intervals would contain the true value.
What Margin of Error Is NOT
- It does NOT measure bias (systematic error).
- It does NOT guarantee the true value is in the interval.
- A biased sample with a small margin of error is still wrong.
Strategy Tips
Tip 1: Larger Sample = Smaller Margin of Error
To reduce the margin of error, increase the sample size.
Tip 2: The Interval Is Centred on the Sample Statistic
Sample proportion ± margin of error gives the interval.
Tip 3: Higher Confidence = Wider Interval
A 99% confidence interval is wider than a 95% one.
Worked Example: Example 1
A poll shows 48% support with a margin of error of ±4%. What is the confidence interval?
to . Interval: 44% to 52%.
Worked Example: SAT-Style
Survey A polls 400 people (margin of error ±5%). Survey B polls 1600 people (margin of error ±2.5%). Which is more precise?
Survey B — larger sample, smaller margin of error.
Worked Example: Example 3
A study estimates the mean height of students is 170 cm ± 3 cm. Can we conclude the mean is exactly 170?
No — we can only say the true mean is likely between 167 and 173 cm.
Practice Problems
Problem 1
A survey finds 35% prefer brand X with margin of error ±3%. Give the confidence interval.
Problem 2
To halve the margin of error, by approximately what factor must you increase the sample size?
Problem 3
Two polls: 52% ± 4% and 48% ± 4%. Do the confidence intervals overlap? What does that imply?
Want to check your answers and get step-by-step solutions?
Common Mistakes
- Thinking margin of error fixes bias. If the sample is biased, the interval is centred on the wrong value.
- Interpreting the confidence level incorrectly. "95% confident" doesn't mean "95% probability the true value is in this interval."
- Thinking bigger sample always means better results. Size reduces random error but not systematic bias.
Key Takeaways
Confidence interval = sample statistic ± margin of error.
Larger sample → smaller margin of error → more precise estimate.
Higher confidence level → wider interval.
Margin of error measures random error, not bias.
If confidence intervals overlap, the difference may not be significant.