Confidence intervals

Definition

Confidence intervals are ranges of values used to estimate a population parameter with a certain level of confidence. They provide an interval within which the true value of the parameter is expected to fall.

5 Must Know Facts For Your Next Test

1. The formula for a confidence interval for a population proportion is given by $\hat{p} \pm Z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$, where $\hat{p}$ is the sample proportion, $Z_{\alpha/2}$ is the critical value from the standard normal distribution, and $n$ is the sample size.
2. A higher confidence level (e.g., 99%) will result in a wider confidence interval than a lower confidence level (e.g., 95%).
3. The margin of error in a confidence interval decreases as the sample size increases.
4. If the population proportion is unknown, it can be estimated using the sample proportion.
5. For large samples, typically $n \geq 30$, the sampling distribution of the sample proportion approximates a normal distribution.

Review Questions

• What happens to the width of a confidence interval if you increase the sample size?
• How do you calculate the margin of error for a population proportion?
• Why does increasing the confidence level make the confidence interval wider?

Related terms

Population Proportion:

The ratio or fraction of members in a population that have a particular attribute.

Margin of Error:

The range above and below the sample statistic in a confidence interval.

Critical Value:

$Z$-score corresponding to a specified level of significance or confidence level in hypothesis testing or estimation.