A confidence interval for a proportion estimates an unknown population percentage using data from a sample. It is widely used in polling, surveys, medical studies, and quality control because we usually cannot measure every person or item in a population. Instead of giving one number, the interval gives a range of plausible values for the true population proportion.
The confidence level describes how reliable the method is over many repeated samples.
Key Facts
- Sample proportion: p-hat = x/n, where x is the number of successes and n is the sample size.
- Confidence interval for a proportion: p-hat ± z*sqrt(p-hat(1 - p-hat)/n).
- Margin of error: ME = z*sqrt(p-hat(1 - p-hat)/n).
- For a 95% confidence interval, z* = 1.96 when the normal model is appropriate.
- Large counts condition: n p-hat >= 10 and n(1 - p-hat) >= 10.
- A 95% confidence level means that about 95% of intervals made by this method would contain the true population proportion.
Vocabulary
- Population proportion
- The true fraction of the entire population that has a certain characteristic.
- Sample proportion
- The fraction of the sample that has a certain characteristic, written as p-hat.
- Confidence interval
- A range of values calculated from sample data that is likely to contain the true population parameter.
- Margin of error
- The amount added to and subtracted from the sample estimate to form a confidence interval.
- Critical value
- A multiplier from the normal distribution, such as z* = 1.96 for 95% confidence, used to set the width of an interval.
Common Mistakes to Avoid
- Saying there is a 95% chance that the specific interval contains the true proportion is wrong because the true proportion is fixed after the interval is computed. The 95% describes the long-run success rate of the method.
- Using p instead of p-hat in the standard error is wrong when the population proportion is unknown. For a one-sample confidence interval, estimate the standard error with p-hat.
- Ignoring the large counts condition is wrong because the normal approximation may be poor for small samples or extreme proportions. Check that n p-hat >= 10 and n(1 - p-hat) >= 10 before using the standard formula.
- Interpreting the interval as a range for individual responses is wrong because the interval estimates a population proportion. It does not predict whether a particular person will answer yes or no.
Practice Questions
- 1 In a poll of 500 voters, 280 support a new policy. Find p-hat and construct a 95% confidence interval for the true proportion of voters who support the policy.
- 2 A survey finds that 72 out of 200 students walk to school. Using z* = 1.96, calculate the margin of error and the 95% confidence interval for the true proportion.
- 3 A poll reports that 48% of voters support Candidate A with a margin of error of 3 percentage points at 95% confidence. Explain why it is not correct to say that there is a 95% chance Candidate A's true support is between 45% and 51%.