Confidence Interval
A confidence interval is a type of estimate computed from the statistics of the observed data. This estimate provides a range of values, which is likely to include an unknown population parameter. The interval has an associated confidence level that quantifies the level of confidence that the parameter lies within the interval. Here's a deeper look into the concept:
Definition and Interpretation
A confidence interval provides an estimate of the margin of error for a sample statistic. For instance, if a poll reports that 40% of people approve of a policy with a 95% confidence interval of ±3%, it means that if the same poll were conducted 100 times, about 95 of those times, the true population proportion would fall within the interval of 37% to 43%.
Historical Context
- The concept of confidence intervals can be traced back to the work of Jerzy Neyman and Egon Pearson in the 1930s. They developed the theory of hypothesis testing and introduced the notion of confidence limits, which laid the groundwork for modern statistical inference.
- Neyman and Pearson's work was influenced by earlier contributions from Karl Pearson, who developed the method of moments and the chi-squared test, but did not explicitly deal with confidence intervals.
Calculation
The calculation of a confidence interval involves:
- Selecting a confidence level, typically 90%, 95%, or 99%.
- Using the sample data to calculate the sample statistic (e.g., sample mean, proportion).
- Using the appropriate standard error or margin of error formula, which depends on the distribution of the data (normal, t-distribution, etc.).
- Constructing the interval using the formula:
CI = statistic ± (critical value * standard error)
Key Points
- The confidence level represents the probability that the estimation procedure produces an interval that includes the true parameter value.
- The width of the confidence interval is influenced by:
- The sample size (larger samples yield narrower intervals).
- The variability in the population (more variability results in wider intervals).
- The confidence level (higher confidence levels result in wider intervals).
Applications
Confidence intervals are widely used in:
- Scientific research for reporting results.
- Political polling to assess public opinion.
- Quality control in manufacturing to set acceptable limits.
- Healthcare to understand the effectiveness of treatments or interventions.
Sources
Related Concepts