Proportional Data Math Example 2

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Example 2

medium
A poll finds p^=0.52\hat{p} = 0.52 supporting a candidate from n=400n=400 voters. Calculate the standard error of p^\hat{p} and construct an approximate 95% confidence interval.

Solution

  1. 1
    Standard error of proportion: SE=p^(1−p^)n=0.52×0.48400=0.2496400=0.000624≈0.025SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.52 \times 0.48}{400}} = \sqrt{\frac{0.2496}{400}} = \sqrt{0.000624} \approx 0.025
  2. 2
    95% CI uses z∗=1.96z^* = 1.96: p^±z∗×SE=0.52±1.96(0.025)=0.52±0.049\hat{p} \pm z^* \times SE = 0.52 \pm 1.96(0.025) = 0.52 \pm 0.049
  3. 3
    Confidence interval: (0.471,0.569)(0.471, 0.569)
  4. 4
    Interpretation: we are 95% confident the true proportion is between 47.1% and 56.9%

Answer

SE≈0.025SE \approx 0.025; 95% CI: (0.471,0.569)(0.471, 0.569). Cannot confirm majority (interval includes values below 50%).
The standard error of a proportion depends on both p^\hat{p} and sample size. The confidence interval extends z∗×SEz^* \times SE on each side of p^\hat{p}. Larger samples produce narrower intervals (more precision).

About Proportional Data

Proportional data expresses quantities as fractions or percentages of a whole, enabling fair comparison across groups of different sizes.

Learn more about Proportional Data →

More Proportional Data Examples

Example 1 easy

In a survey, 45 out of 180 students prefer online learning. Calculate the sample proportion [formula

Example 3 easy

In a class of 25 students, 15 passed the test. Calculate [formula] and find the number expected to p

Example 4 hard

Compare two proportions: Group A: 30/120 support policy. Group B: 45/150 support policy. Test if the

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