A 95% confidence interval means that if we took many samples, approximately 95% of them would contain the true population parameter.
The width of a confidence interval depends on the sample size and variability; larger samples give narrower intervals.
Confidence intervals for proportions are often calculated using the formula: $\hat{p} \pm Z \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ where $\hat{p}$ is the sample proportion, $Z$ is the z-score associated with the desired confidence level, and $n$ is the sample size.
The margin of error increases as the confidence level increases, meaning a wider interval for higher confidence levels.
If assumptions (e.g., normality) are not met, alternative methods like bootstrapping might be used to construct confidence intervals.
Review Questions
What does a 95% confidence interval imply about repeated sampling?
How does sample size affect the width of a confidence interval?
Write down and explain the formula for constructing a confidence interval for a population proportion.
Related terms
Margin of Error: The amount added and subtracted from a point estimate to create a confidence interval.
Sample Proportion: $\hat{p}$: The ratio of individuals in a sample with a particular characteristic to the total number of individuals in that sample.
Z-Score: A statistical measure that describes a value's relationship to the mean of a group of values, often used in calculating confidence intervals.