Confidence interval

Definition

A confidence interval is a statistical range, derived from a data set, that is likely to contain the true value of an unknown population parameter. It provides an estimated interval around a sample mean or proportion, along with a confidence level that quantifies the level of certainty about this estimation. The concept is closely related to probability distributions, as it relies on the principles of sampling distributions and the normal distribution to make inferences about population parameters.

5 Must Know Facts For Your Next Test

1. Confidence intervals are typically expressed with a specific confidence level, such as 95% or 99%, indicating the probability that the interval contains the true population parameter.
2. The width of a confidence interval depends on the sample size and variability; larger samples tend to yield narrower intervals, providing more precise estimates.
3. Confidence intervals can be calculated for various statistics, including means, proportions, and differences between groups.
4. A 95% confidence interval means that if the same population were sampled multiple times, approximately 95% of those intervals would contain the true population parameter.
5. Confidence intervals help to quantify uncertainty in estimates and are widely used in inferential statistics to guide decision-making.

Review Questions

• How does the sample size influence the width of a confidence interval?
  • The sample size has a significant impact on the width of a confidence interval. As the sample size increases, the standard error decreases, resulting in a narrower confidence interval. This means that larger samples provide more precise estimates of population parameters and reduce uncertainty. In contrast, smaller samples often lead to wider intervals, reflecting greater variability and less certainty about where the true parameter lies.
• Discuss how confidence intervals can be utilized in hypothesis testing to make inferences about population parameters.
  • In hypothesis testing, confidence intervals serve as a tool for assessing whether sample data supports or contradicts a null hypothesis. By constructing a confidence interval around an estimate, researchers can see if the value posited by the null hypothesis falls within this interval. If it does not, this suggests evidence against the null hypothesis. This method combines estimation with significance testing, enabling researchers to make informed conclusions based on their data.
• Evaluate the implications of selecting different confidence levels when calculating confidence intervals and how this affects decision-making in research.
  • Choosing different confidence levels when calculating confidence intervals can significantly influence research findings and decisions. A higher confidence level (e.g., 99%) yields wider intervals, which may reduce precision but increases certainty about capturing the true parameter. Conversely, opting for a lower confidence level (e.g., 90%) results in narrower intervals but less certainty. This trade-off affects how confidently researchers can make claims based on their data and may lead to different conclusions regarding hypotheses or policy decisions.

Related terms

margin of error: The margin of error indicates the amount of random sampling error in a survey's results, reflecting the potential difference between the sample estimate and the actual population value.

sample size:

Sample size refers to the number of observations or data points collected from a population, which directly affects the reliability and precision of estimates, including confidence intervals.

hypothesis testing: Hypothesis testing is a statistical method used to determine whether there is enough evidence to reject a null hypothesis, often using confidence intervals to evaluate the validity of claims.