8 Confidence Interval
A confidence interval (CI) is a statistical tool used to estimate a range of values within which a population parameter is likely to fall, based on sample data. Rather than providing a single point estimate, a confidence interval expresses the uncertainty inherent in sampling, giving both a lower and an upper bound for the parameter of interest.
Confidence intervals are typically expressed with a confidence level, such as 90%, 95%, or 99%. For example, a 95% confidence interval for a population mean suggests that if we were to take 100 different samples and calculate the CI for each, approximately 95 of these intervals would contain the true population mean.
Key elements in constructing a confidence interval include:
- Point estimate: The sample statistic used to estimate the population parameter (e.g., sample mean). - Standard error (SE): Measures the variability of the point estimate across different samples.
- Critical value: Corresponds to the desired confidence level and is derived from the probability distribution of the estimator (commonly the Z-distribution for large samples or the t-distribution for small samples).
Confidence intervals (Figure 8.1) are closely related to Central Tendency and Statistical Dispersion. The point estimate often represents a measure of central tendency, while the width of the interval depends on the variability of the data. Narrower intervals indicate more precise estimates, whereas wider intervals reflect greater uncertainty.
Graphical representation of confidence intervals, such as error bars in charts, can help visualize the range of plausible values and the degree of uncertainty associated with the estimate. Understanding and interpreting confidence intervals is essential for making informed decisions and drawing reliable conclusions in research and applied statistics khan_academy_ci?, statisticsbyjim_ci?, geeksforgeeks_ci?.