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Confidence Interval for One Proportion

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Confidence Interval for One Proportion: Definition, Formula, and Interpretation

Confidence Interval for One Proportion

Definition: A confidence interval for a proportion provides a range of values that is likely to contain the true population proportion with a certain level of confidence.

Formula:p^±zα/2p^(1−p^)n\hat p \pm z_{\alpha/2} \sqrt{\frac{\hat p(1-\hat p)}{n}}

Where:

  • p^\hat p is the sample proportion
  • zα/2z_{\alpha/2} is the critical value from the standard normal distribution
  • nn is the sample size
Example:

Suppose in a sample of 200 people, 30 are left-handed. To calculate a 95% confidence interval:

0.15±1.960.15(1−0.15)200≈(0.10,0.20)0.15 \pm 1.96 \sqrt{\frac{0.15(1-0.15)}{200}} \approx (0.10, 0.20)

We can be 95% confident that the true proportion of left-handed people in the population is between 10% and 20%.

Interpretation and Assumptions

Interpretation: A 95% confidence interval means that if we repeated this sampling process many times, about 95% of the intervals would contain the true population proportion.

Assumptions:

  1. The sample is randomly selected from the population.
  2. Each observation in the sample is independent of the others.
  3. The sample size is large enough (both np and n(1-p) should be greater than or equal to 10).
  4. The population size is much larger than the sample size (at least 10 times larger).

If these assumptions are violated, consider using alternative methods such as exact binomial intervals or bootstrapping.

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