Integrated Math 3 / Unit 8: Statistics and Inference
8.5 Margin of Error and Confidence
Notes
A sample proportion $\hat{p}$ estimates the true population proportion, but samples vary.
- The margin of error (ME) tells how far the estimate may be from the truth. A rough 95% rule is $\displaystyle ME \approx \frac{1}{\sqrt{n}}$.
- The estimate gives an interval: $\hat{p} \pm ME$.
- A bigger sample gives a smaller margin of error. To cut the margin of error in half, you need 4 times the sample size.
- If an interval includes 50%, a poll cannot say that either side has a majority.
Practice
A poll of 400 randomly chosen people has a margin of error of about $\displaystyle \frac{1}{\sqrt{n}}$. What is it?
$\displaystyle \frac{1}{\sqrt{400}} = \frac{1}{20} = 0.05$.
A poll finds 54% support for a proposal, with a margin of error of 3%. What interval is likely to contain the true percent?
$54\% \pm 3\%$ gives 51% to 57%.
A poll finds 52% support for a candidate with a margin of error of 3%. What can be concluded?
The true support could be as low as 49%.
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