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.
Understanding Statistics: Confidence Interval for a Proportion
The calculation starts by treating each observation as one of two outcomes, such as supports a policy or does not support it. The share found in the sample is the best single estimate available, but samples vary by chance. A different randomly chosen group would usually produce a somewhat different share.
The margin of error measures this expected sampling variation. It depends on the estimated share, the number of observations, and the chosen confidence level.
The value used for the confidence level comes from the normal distribution. Higher confidence requires a wider interval because the method must catch the true value more often over repeated sampling.
Sample size has a strong effect on precision. If a survey grows from 400 people to 1600 people, the margin of error is roughly cut in half, not cut to one quarter. This happens because random variation shrinks with the square root of the sample size.
Collecting many more responses can help, though it can be expensive or slow. Proportions near one half have the most possible random variation.
Proportions close to zero or one tend to have less variation. This is why planners often use one half when estimating the largest sample size they might need before a survey begins.
The usual interval method needs enough successes and enough non-successes in the data. A sample with only a few people who have the characteristic can produce a lopsided distribution that the normal model does not describe well. Checking that both expected counts are at least about ten is a practical safeguard.
Independence matters too. Randomly selecting people is important, and sampling without replacement should usually involve no more than one tenth of the full population. When samples are small or the proportion is very close to zero or one, statisticians may use methods such as the Wilson interval instead of the basic normal interval.
A confidence interval only addresses chance variation from sampling. It cannot repair a biased sample. For example, an online poll may have thousands of responses, yet it can still miss people without internet access or attract unusually motivated voters.
Leading wording, skipped responses, dishonest answers, and recording mistakes can shift results outside the reported range. In class problems, state the population clearly, identify what counts as a success, and check the sampling conditions before calculating.
Interpret the final range as an estimate for the population percentage, not as a range containing a fixed percentage of individual people. A 95 percent interval does not mean there is a 95 percent chance that the already fixed population proportion lies inside this particular interval.
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%.