95% confidence interval

Definition

A 95% confidence interval is a statistical range that is likely to contain the true population parameter 95% of the time based on sample data. This concept helps quantify the uncertainty associated with sample estimates and provides a way to express how confident we are in our results. A wider interval indicates more uncertainty about the parameter, while a narrower interval suggests more precision.

5 Must Know Facts For Your Next Test

1. The 95% confidence interval is typically calculated using the formula: Sample Mean ± (Critical Value × Standard Error), where the critical value corresponds to a 95% confidence level.
2. In many cases, a normal distribution is assumed for the sample mean when calculating a confidence interval, especially when the sample size is large due to the Central Limit Theorem.
3. Confidence intervals can be constructed for various population parameters such as means, proportions, and regression coefficients.
4. If you were to take many samples and create a confidence interval from each one, approximately 95% of those intervals would contain the true population parameter.
5. The width of a 95% confidence interval can be affected by sample size; larger samples tend to produce narrower intervals, indicating greater precision in estimating the population parameter.

Review Questions

• How does increasing the sample size impact the width of a 95% confidence interval, and why is this important?
  • Increasing the sample size generally leads to a narrower 95% confidence interval. This happens because larger samples reduce the standard error, which is part of the formula used to calculate the interval. A narrower interval means that we have more precise estimates of the population parameter, which is crucial for making informed decisions based on statistical analysis.
• Discuss how the choice of confidence level affects the width of a confidence interval and its interpretation.
  • Choosing a higher confidence level, such as 99%, will result in a wider confidence interval compared to a 95% level. This wider interval reflects greater uncertainty regarding where the true population parameter lies. While a higher confidence level gives us more assurance that our interval contains the true parameter, it can also lead to less precise estimates, making it important to balance confidence with practical utility.
• Evaluate how confidence intervals can provide insights into statistical significance and decision-making in hypothesis testing.
  • Confidence intervals play a significant role in hypothesis testing by providing a range of plausible values for population parameters. If a 95% confidence interval for a mean does not include the null hypothesis value (often zero), it suggests that the result is statistically significant at that level. This relationship helps researchers make decisions about hypotheses based on empirical data, as it indicates whether observed effects are likely due to random chance or represent real differences in populations.

Related terms

Sample Mean: The average value calculated from a sample of data, which serves as an estimate of the population mean.

Margin of Error: The amount of error allowed in the estimation of the population parameter, often used to calculate confidence intervals.

Standard Deviation: A measure of the amount of variation or dispersion in a set of values, crucial for calculating confidence intervals.