# Confidence intervals for one population proportion

The proportion of patients with a specific disease who are under a new drug. The proportion of graduating high school students who can read at the eighth grade level. The proportion of Republicans who will vote for Bush. The proportion of Democrats who will vote for Bush. Estimates based on samples are, of course, subject to sampling error random errorand it is important to evaluate the precision of our estimates. In public health this is most commonly done by computing a confidence interval.

One can compute confidence intervals all types of estimates, but this short module will provide the conceptual background for computing confidence intervals and will then focus on the computation and interpretation of confidence intervals for a mean or a proportion in a single group.

## Confidence Interval for the Population Proportion

This is particularly relevant for the analysis and presentation of descriptive studies, such as a case series, in which one is simply trying to accurately report characteristics of a single group.

Later modules will address the computation and interpretation of confidence intervals for estimates from analytical studies e. How do I gauge the precision of an estimated mean or an estimated proportion in a single sample?

How do I interpret and calculate a confidence interval for an estimate in a single sample? After successfully completing this unit, the student will be able to: Explain what a confidence interval is. Interpret the confidence interval for a mean or a proportion from a single group.

Use R to compute a confidence interval for the mean in a single group Use R to compute a confidence interval for a proportion in a single group Estimating Population Parameters in a Single Group The goal of exploratory or descriptive studies is not to formally compare groups in order to test for associations between exposures and health outcomes, but to estimate and summarize the characteristics of a particular population of interest.

Typical examples would be a case series of humans who had been diagnosed and treated for bird flu or a cross-sectional study in a community for the purpose of better understanding the current health status and potential challenges for the future.

The variables being estimated would logically include both continuous variables e. For both continuous variables e. Recall that sample means and sample proportions are unbiased estimates of the corresponding population parameters.

Confidence Intervals For both continuous and dichotomous variables, the confidence interval estimate CI is a range of likely values for the population parameter based on: In practice, however, we select one random sample and generate one confidence interval, which may or may not contain the true mean.

Key Concept A confidence interval does not reflect the variability in the unknown parameter. Rather, it reflects the amount of random error in the sample and provides a range of values that are likely to include the unknown parameter.

Another way of thinking about a confidence interval is that it is the range of likely values of the parameter with a specified level of confidence which is similar to a probability.

The Central Limit Theorem states that, for large samples, the distribution of the sample means is approximately normally distributed with a mean: There is often confusion regarding standard deviations and standard errors. Standard deviations describe variability in a measure among experimental units e.

Standard errors represent variability in estimates of a mean or proportion; i. Another way to think of this is that standard deviations describe the variability in a population while standard errors represent variability in the sampling means or proportions.In statistics, a binomial proportion confidence interval is a confidence interval for the probability of success calculated from the outcome of a series of success–failure experiments (Bernoulli trials).In other words, a binomial proportion confidence interval is an interval estimate of a success probability p when only the number of experiments n and the number of successes n S are known.

One can compute confidence intervals all types of estimates, but this short module will provide the conceptual background for computing confidence intervals and will then focus on the computation and interpretation of confidence intervals for a mean or a proportion in a single group.

The result is called a confidence interval for the population proportion, p. The formula for a CI for a population proportion is is the sample proportion, n is the sample size, and z* is the appropriate value from the standard normal distribution for your desired confidence level.

Confidence Intervals for One Proportion Introduction This routine calculates the sample size necessary to achieve a specified interval width or distance from the sample proportion to the confidence limit at a stated confidence level for a confidence interval for one proportion.

Since the above requirements are satisfied, we can use the following four-step approach to construct a confidence interval. Identify a sample statistic. Since we are trying to estimate a population proportion, we choose the sample proportion () as the sample statistic.

Select a confidence level. for 95% of all possible samples, the sample proportion will be within two standard errors of the true population proportion.

Thus, a 68% confidence interval for the percent of all Centre Country households that don't meet the EPA guidelines is given by. % ± %.

Confidence interval for a proportion | Sample Size Calculators