125 Estimating a Population Proportion (1 of 3)
Learning Objectives
- Construct a confidence interval to estimate a population proportion when conditions are met. Interpret the confidence interval in context.
Introduction
In “Estimating a Population Proportion,” we continue our discussion of estimating a population proportion with a confidence interval. Recall that the purpose of a confidence interval is to use a sample proportion to construct an interval of values that we can be reasonably confident contains the true population proportion.
The basic idea is summarized here:
- When we select a random sample from the population of interest, we expect the sample proportion to be a good estimate of the population proportion. But we also know that sample proportions vary, so we expect some error. (Remember that the error here is due to chance. It is not due to a mistake that anyone made.)
- For a given sample proportion, we will not know the amount of error, so we use the standard error as an estimate for the average amount of error we expect in sample proportions. (Recall that the standard error is the expected standard deviation of sample proportions when we take many, many random samples.)
- If a normal model is a good fit for the sampling distribution, then about 95% of sample proportions estimate the population proportion within 2 standard errors. We say that we are 95% confident that the following interval contains the population proportion.
[latex]\begin{array}{l}p\text{}±\text{}\mathrm{margin}\text{}\mathrm{of}\text{}\mathrm{error}\\ p\text{}±\text{}2(\mathrm{standard}\text{}\mathrm{error})\\ p\text{}±\text{}2\sqrt{\frac{p(1-p)}{n}}\end{array}[/latex]
You may realize that this formula for the confidence interval is a bit odd, since our goal in calculating the confidence interval is to estimate the population proportion p. Yet the formula requires that we know p. In the section “Introduction to Statistical Inference,” we used an estimate for p from a previous study when calculating the confidence interval. This is not the usual way statisticians estimate the standard error, but it captured the main idea and allowed us to practice finding and interpreting confidence intervals. Now, we develop a different way to estimate standard error that is commonly used in statistical practice.
Example
Community College Students and Gender
According to a 2010 report from the American Council on Education, females make up 57% of the college population in the United States. Students in a statistics class at Tallahassee Community College want to determine the proportion of female students at TCC. They select a random sample of 135 TCC students and find that 72 are female, which is a sample proportion of 72 / 135 ≈ 0.533. So 53.3% of the students in the sample are female.
What can they conclude about the proportion of females at the college? How confident can they be in their estimate?
To answer these questions, we need to find a confidence interval.
Checking co