Interval estimation

Interval estimation is a type of estimation in statistics that is used to specify a range of numbers within which a population parameter is estimated to lie. Unlike point estimation, which provides a single value as an estimate of a parameter, interval estimation gives a range of plausible values for the parameter, thereby offering a measure of estimation uncertainty. This range is called a confidence interval (CI), and it is constructed so that, with a certain level of confidence, it contains the true parameter value.

Basics of Interval Estimation

Interval estimation revolves around the concept of capturing the true parameter value within a specified range, with a certain degree of confidence. This confidence level, often expressed as a percentage (e.g., 95% or 99%), indicates the proportion of intervals that, if constructed from many independent samples, would contain the true parameter. The actual calculation of a confidence interval depends on several factors, including the sample size, the variability in the data, and the desired confidence level.

The general form of a confidence interval for a population mean, when the population standard deviation is known, is given by:

\[ \bar{x} \pm Z \left( \frac{\sigma}{\sqrt{n}} \right) \]

where:

• \( \bar{x} \) is the sample mean,
• \( Z \) is the Z-score associated with the desired confidence level,
• \( \sigma \) is the population standard deviation, and
• \( n \) is the sample size.

For situations where the population standard deviation is unknown, the t-distribution is used instead of the normal distribution, and the formula adjusts to incorporate the sample standard deviation \( s \) in place of \( \sigma \).

Types of Confidence Intervals

There are various types of confidence intervals, tailored for different statistical parameters and situations. Some common types include:

• Confidence intervals for the mean (when population variance is known or unknown),
• Confidence intervals for proportions,
• Confidence intervals for the difference between two means or two proportions,
• Confidence intervals for variance and standard deviation.

Importance of Interval Estimation

Interval estimation is crucial in statistical inference because it provides more information than a simple point estimate. By offering a range of possible values, it helps researchers assess the reliability and precision of their estimates. This is particularly important in fields such as medicine, biology, and economics, where decisions often rely on understanding the uncertainty associated with estimated parameters.

Limitations

While interval estimation is a powerful tool, it has limitations. The confidence level does not describe the probability that the specific interval calculated from a sample contains the true parameter value. Instead, it describes the long-run proportion of such intervals that would contain the parameter if the study were repeated numerous times. Additionally, confidence intervals are based on assumptions about the sample and the population that may not always hold true, potentially leading to misleading conclusions.

See Also

• Point estimation
• Statistical inference
• Margin of error
• Sampling (statistics)
• Standard error