Confidence Intervals for Proportions

Start with a random sample of size n. Count the number of successes x (the outcomes with the characteristic of interest). The sample proportion is p̂ = x/n. This is your point estimate of the unknown population proportion p. For example, if 42 out of 100 surveyed students own a pet, then p̂ = 0.42.

Three conditions must be met before constructing the interval. (1) Random: the sample must be a simple random sample or come from a randomized experiment. (2) Normal (Large Counts): both np̂ ≥ 10 and n(1 − p̂) ≥ 10, ensuring the sampling distribution of p̂ is approximately Normal. (3) Independence: either sampling with replacement, or the sample size n is less than 10% of the population size (the 10% condition). Skipping these checks can invalidate the interval.

The critical value z comes from the standard Normal distribution and corresponds to your chosen confidence level C. Common values: for 90% confidence, z = 1.645; for 95% confidence, z = 1.960; for 99% confidence, z = 2.576. You find z such that the area between −z and z under the standard Normal curve equals C. In other words, z = invNorm((1 + C)/2).

The standard error (SE) of p̂ measures how much p̂ varies from sample to sample. It is SE = √(p̂(1 − p̂)/n). The margin of error (ME) is ME = z × SE = z × √(p̂(1 − p̂)/n). A larger sample size n decreases the margin of error, producing a more precise interval. A higher confidence level increases z*, which widens the interval.

The one-sample z-interval for a proportion is: (p̂ − ME, p̂ + ME). Write your conclusion in context: 'We are C% confident that the true proportion of [population] who [characteristic] is between [lower bound] and [upper bound].' Always reference the population, the variable, and the confidence level in your interpretation. Never say 'there is a C% probability that p is in this interval' — p is a fixed (though unknown) number, not a random variable.

• \hat{p} — Sample proportion (dimensionless): The proportion of successes observed in the sample; equals x/n where x is the count of successes.
• z^* — Critical value (dimensionless): The number of standard deviations from the mean corresponding to the desired confidence level in the standard Normal distribution (e.g., 1.96 for 95% confidence).
• n — Sample size (count): The total number of individuals or observations in the sample. Larger n produces a narrower interval.
• \sqrt{\hat{p}(1-\hat{p})/n} — Standard error (dimensionless): Estimates the standard deviation of the sampling distribution of p̂, measuring the typical distance between p̂ and the true p.

Problem. A school administrator surveys a random sample of 200 students and finds that 74 of them walk to school. Construct a 95% confidence interval for the true proportion of all students at this school who walk to school.

Answer. 95% CI for p: (0.303, 0.437)

• Sampling Distributions for Proportions — The confidence interval formula relies on the sampling distribution of p̂ being approximately Normal, which requires understanding how p̂ varies across repeated samples.
• Significance Tests for Proportions — Hypothesis tests and confidence intervals are complementary inference tools; a value outside the CI corresponds to rejecting it as a null hypothesis value at the matching significance level.
• Margin of Error and Sample Size Determination — Choosing sample size in advance to achieve a desired margin of error is a direct application of the confidence interval formula solved for n.
• Confidence Intervals for Means — The logic of construction and interpretation is identical for means, but the standard error formula and the t-distribution replace the Normal-based approach used for proportions.