Margin of Error Math Example 1

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Example 1

easy
A poll of n=400n=400 voters finds p^=0.55\hat{p}=0.55 supporting a candidate. Calculate the margin of error at 95% confidence and interpret the result.

Solution

  1. 1
    SE=p^(1−p^)n=0.55×0.45400=0.2475400=0.000619≈0.0249SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.55 \times 0.45}{400}} = \sqrt{\frac{0.2475}{400}} = \sqrt{0.000619} \approx 0.0249
  2. 2
    E=z∗×SE=1.96×0.0249≈0.049≈±5%E = z^* \times SE = 1.96 \times 0.0249 \approx 0.049 \approx \pm 5\%
  3. 3
    Confidence interval: 0.55±0.049=(0.501,0.599)0.55 \pm 0.049 = (0.501, 0.599)
  4. 4
    Interpretation: we are 95% confident between 50.1% and 59.9% of voters support the candidate

Answer

E≈±5%E \approx \pm 5\%; 95% CI: (50.1%,59.9%)(50.1\%, 59.9\%). Majority support is likely.
Margin of error quantifies polling uncertainty. 'Candidate leads 55% ± 5%' means the true support could be anywhere from 50% to 60%. A margin of error that doesn't include 50% still suggests the candidate leads, but we can't be certain.

About Margin of Error

The maximum expected difference between the sample statistic and the true population parameter; it is half the width of a confidence interval.

Learn more about Margin of Error →

More Margin of Error Examples

Example 2 medium

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Example 3 easy

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Example 4 hard

Two polls: Poll A ([formula]): margin of error ±10%. Poll B ([formula]): margin of error ±2.5%. By w

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