Confidence Intervals
| Title |
|---|
| Confidence interval of a mean |
| Confidence interval of a proportion |
| Connfidence interval of a variance |
Introduction to Estimation
In statistics, estimation is a pivotal concept aimed at inferring the characteristics of a population by examining a subset of its members, or a sample. There are two primary forms of statistical estimates:
Point Estimates
This is a single statistic, represented as \(\hat{\theta}\), derived from sample data to estimate a population parameter \(\theta\).
For instance, the sample mean \[\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i\] is used as a point estimate of the population mean \(\mu\).
While point estimates provide a specific value as an estimate, they lack information on the accuracy of this estimate.
Interval Estimates
These offer a range of values, calculated from the sample data, within which the population parameter is expected to lie. This range is associated with a confidence level, such as \(95\%\) or \(99\%\), indicating the degree of certainty or confidence we have that the interval indeed contains the parameter.
Confidence intervals (CI) are a type of interval estimate characterized by their ability to quantify uncertainty. A confidence interval is mathematically expressed as \[[ \hat{\theta}_L, \hat{\theta}_U ],\] where: \(\hat{\theta}_L\) and \(\hat{\theta}_U\) are the lower and upper bounds of the interval, respectively.
The confidence level, often denoted as \((1 - \alpha) \times 100\%\), where \(\alpha\) is the significance level, provides the probability that the interval \([ \hat{\theta}_L, \hat{\theta}_U ]\) contains the population parameter \(\theta\). For example, a 95% confidence interval suggests that if we were to draw 100 different samples and calculate a CI for each, approximately 95 of those intervals would be expected to encompass\(\theta\).
Formally, if \(x_1, \cdots, x_n\) represent the observed data and \(X_1, \cdots, X_n\) denote the corresponding random variables (the underlying sample), the confidence interval offers a probabilistic assessment of where the true population parameter \(\theta\) lies based on the sample data.
The application of confidence intervals goes beyond mere point estimation by providing a measure of the estimate’s reliability. Through the subsequent sections, we will explore the computation, interpretation, and practical applications of confidence intervals in detail.