Confidence intervals

Definition

Confidence intervals provide a range of values that likely contain the true population parameter. They quantify the uncertainty of an estimate.

5 Must Know Facts For Your Next Test

1. A 95% confidence interval means that if we took many samples, approximately 95% of them would contain the true population parameter.
2. The width of a confidence interval depends on the sample size and variability; larger samples give narrower intervals.
3. 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.
4. The margin of error increases as the confidence level increases, meaning a wider interval for higher confidence levels.
5. 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.