Confidence Interval (CI): Statistical Range Estimation

A Confidence Interval (CI) is a range of values derived from sample data that is likely to contain a population parameter with a certain level of confidence.

A Confidence Interval (CI) is a statistical concept that represents a range of values, derived from sample data, which is likely to contain the true population parameter with a specified level of confidence. Confidence intervals provide an estimated range of where we expect the true measure (such as mean or proportion) to lie, given the variation inherent in the data.

Definition and Key Concepts

Confidence interval is defined mathematically as:

$$ CI = \hat{\theta} \pm Z_{\frac{\alpha}{2}} \left(\frac{\sigma}{\sqrt{n}}\right) $$

Where:

• \( \hat{\theta} \) is the sample statistic (e.g., sample mean)
• \( Z_{\frac{\alpha}{2}} \) is the Z-value from the standard normal distribution for the desired confidence level
• \( \sigma \) is the population standard deviation
• \( n \) is the sample size

Types of Confidence Intervals

• Mean Confidence Interval: Used for estimating the population mean.
• Proportion Confidence Interval: Used for estimating the population proportion.
• Variance Confidence Interval: For estimating population variance.
• Regression Confidence Interval: For estimating parameters in regression analysis.

Special Considerations

Confidence Levels

The confidence level is the probability that the confidence interval contains the true population parameter. Common confidence levels are 90%, 95%, and 99%, leading to different Z-values:

• 90% confidence level: \( Z = 1.645 \)
• 95% confidence level: \( Z = 1.96 \)
• 99% confidence level: \( Z = 2.576 \)

Margin of Error

The margin of error represents the extent of the range around the sample statistic within which the population parameter is likely to fall. It reflects the precision of the estimation.

Examples

Example 1: Mean Confidence Interval

Suppose we have a sample mean (\( \bar{x} \)) of 50, a known population standard deviation (\( \sigma \)) of 10, and a sample size (\( n \)) of 100. For a 95% confidence level:

$$ \text{Margin of Error} = Z_{\frac{\alpha}{2}} \cdot \left(\frac{\sigma}{\sqrt{n}}\right) = 1.96 \cdot \left(\frac{10}{\sqrt{100}}\right) = 1.96 $$

$$ CI = 50 \pm 1.96 = (48.04, 51.96) $$

Example 2: Proportion Confidence Interval

If 60 out of 100 surveyed people prefer a new product, the sample proportion \( \hat{p} \) is 0.60. For a 95% confidence level:

$$ \text{Margin of Error} = Z_{\frac{\alpha}{2}} \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = 1.96 \cdot \sqrt{\frac{0.60 \times 0.40}{100}} \approx 0.096 $$

$$ CI = 0.60 \pm 0.096 = (0.504, 0.696) $$

Historical Context

The concept of confidence intervals was introduced by Jerzy Neyman in 1937 as a part of the Neyman-Pearson framework of statistical hypothesis testing. Neyman’s work on confidence intervals paved the way for modern inferential statistics.

Applicability

Confidence intervals are widely used in various fields such as:

• Medicine: For estimating treatment effects
• Economics: For predicting economic indicators
• Engineering: For quality control and reliability testing
• Social Sciences: For survey results analysis

Comparisons to Related Terms

• Point Estimate: A single value derived from sample data used to estimate a population parameter, as opposed to the range provided by a confidence interval.
• Hypothesis Testing: Uses confidence intervals to make inferences about population parameters based on sample data.

FAQs

Why do we use confidence intervals?

Confidence intervals provide a range of plausible values for the population parameter and reflect the uncertainty inherent in any statistical estimate.

Do larger samples provide more accurate confidence intervals?

Yes, larger samples tend to produce narrower confidence intervals, indicating more precise estimates of the population parameter.

How is the confidence level selected?

The confidence level is typically chosen based on the context of the study and the acceptable risk of error (e.g., 95% for general scientific studies).

References

• Neyman, J. (1937). “Outline of a Theory of Statistical Estimation Based on the Classical Theory of Probability”. Philosophical Transactions of the Royal Society of London, Vol. 236, pp. 333–380.
• Moore, D.S., McCabe, G.P., & Craig, B.A. (2021). Introduction to the Practice of Statistics, 10th Edition. W.H. Freeman and Company.

Summary

A Confidence Interval (CI) is a crucial statistical tool for estimating the range within which a population parameter lies, based on sample data. It incorporates the level of confidence and margin of error to provide an understanding of the probable accuracy of the estimate, making it invaluable in various fields of research and application.