A confidence interval (CI) is a range of values, derived from sample data, that is likely to contain the value of an unknown population parameter. The confidence interval provides two possible values—an upper and a lower bound—accompanied by a confidence level which represents the probability that the interval will contain the population parameter. Commonly used confidence levels are 90%, 95%, and 99%.
Mathematical Representation
A confidence interval can be generally expressed as:
\hat{\theta} \pm ME
Where:
- \(\hat{\theta}\) = Sample estimate of the population parameter
- \(ME\) = Margin of error
The margin of error depends on the standard error of the estimate and the critical value from the Z or t-distribution, corresponding to the desired confidence level.
Importance of Confidence Intervals
Confidence intervals are essential in hypothesis testing, determining the reliability of estimates, and guiding decision-making processes. They provide a range instead of a point estimate, which helps in understanding the plausible values a population parameter might take.
Calculating Confidence Intervals
For Population Mean
When estimating the population mean (\(\mu\)), and the sample size (\(n\)) is large (\(n > 30\)), the confidence interval can be calculated using the Z-distribution:
Where:
- \(\bar{x}\) = Sample mean
- \(Z_{\alpha/2}\) = Critical value from the standard normal distribution
- \(\sigma\) = Population standard deviation (or sample standard deviation for large samples)
- \(n\) = Sample size
For Population Proportion
When estimating a population proportion \( p \), the confidence interval is calculated as:
Where:
- \(\hat{p}\) = Sample proportion
Practical Examples
Example 1: Population Mean
A sample of 50 students’ test scores yields a mean score of 78 with a known standard deviation of 10. To find the 95% confidence interval for the population mean score:
Example 2: Population Proportion
A survey finds that 60% of a sample of 200 people favor a new policy. To find the 95% confidence interval for the proportion:
Special Considerations
- Sample Size: As the sample size decreases, the margin of error increases, leading to a wider confidence interval.
- Confidence Level: Higher confidence levels (e.g., 99%) result in wider intervals.
- Distribution: For smaller sample sizes, especially when estimating the mean, the t-distribution is used instead of the Z-distribution.
Historical Context
The concept of confidence intervals was introduced by Jerzy Neyman in 1937, providing a framework for statistical inference, utilizing probability to measure the reliability of an estimate.
Related Terms
- Point Estimate: A single value estimate of a population parameter (e.g., sample mean \(\bar{x}\)).
- Margin of Error: The amount by which the sample estimate is expected to vary from the true population parameter.
- Standard Error: The standard deviation of the sampling distribution of a statistic.
FAQs
Why is a 95% confidence interval most commonly used?
What happens if the assumptions of a confidence interval are violated?
References
- Neyman, J. (1934). “On the two different aspects of the representative method: The method of stratified sampling and the method of purposive selection”. Journal of the Royal Statistical Society.
- Freedman, D., Pisani, R., & Purves, R. (2007). Statistics. W. W. Norton & Company.
Summary
Confidence intervals are a foundational concept in statistics, allowing estimates from sample data to be placed within a range that likely contains the population parameter. Relying on probabilistic measures, confidence intervals aid in understanding the reliability and accuracy of statistical estimates.
