A sample proportion, written as p-hat, is the fraction of a sample that has a particular characteristic. If you take many random samples from the same population, the value of p-hat will change from sample to sample. The sampling distribution of p-hat describes this variation and helps us predict how close a sample result is likely to be to the true population proportion.
This idea is central to polls, quality control, medical studies, and any situation where a percentage is estimated from data.
When the sample is random, independent, and large enough, the sampling distribution of p-hat is approximately normal. Its center is the true population proportion p, and its spread is measured by the standard error sqrt(p(1 - p)/n). Larger samples make the distribution narrower because random variation is reduced.
The normal model allows students to compute probabilities and build confidence intervals using z-scores.
Key Facts
- Sample proportion: p-hat = x/n, where x is the number of successes and n is the sample size.
- Mean of the sampling distribution: mu_p-hat = p.
- Standard error of p-hat: sigma_p-hat = sqrt(p(1 - p)/n).
- Normal approximation condition: np >= 10 and n(1 - p) >= 10.
- Independence condition: observations should be independent, often checked with n <= 0.10N when sampling without replacement.
- Z-score for a sample proportion: z = (p-hat - p)/sqrt(p(1 - p)/n).
Vocabulary
- Sample proportion
- The fraction of individuals in a sample that have the characteristic being studied.
- Sampling distribution
- The probability distribution of a statistic over many random samples of the same size from the same population.
- Population proportion
- The true fraction p of the entire population that has the characteristic of interest.
- Standard error
- The standard deviation of a sampling distribution, measuring how much a statistic typically varies from sample to sample.
- Normal approximation
- The use of a bell-shaped normal model to describe a sampling distribution when sample size conditions are met.
Common Mistakes to Avoid
- Using p-hat instead of p in the standard error formula for a probability calculation. The sampling distribution under a claimed population proportion uses sigma_p-hat = sqrt(p(1 - p)/n).
- Forgetting to check np >= 10 and n(1 - p) >= 10. Without enough expected successes and failures, the sampling distribution may be skewed and the normal model may be inaccurate.
- Thinking that p-hat is always equal to p. A sample proportion is a random statistic, so it varies from sample to sample even when the population proportion stays fixed.
- Ignoring independence when sampling without replacement. If the sample is more than 10% of the population, the standard error formula may overstate or understate the true variability.
Practice Questions
- 1 A population has p = 0.40, and random samples of size n = 100 are taken. Find the mean and standard error of the sampling distribution of p-hat.
- 2 A company claims that 8% of its products are defective. For a random sample of n = 200 products, find the probability that p-hat is greater than 0.11 using the normal approximation.
- 3 A survey takes a random sample of 25 people from a population where p = 0.04. Explain whether a normal model for the sampling distribution of p-hat is appropriate, and justify your answer using the success-failure condition.