20.6 Estimation

Estimation is a fundamental statistical technique used to infer characteristics of an entire population based on data collected from a smaller, representative sample. It allows us to make educated guesses about unknown population parameters, such as the mean, proportion, or variance, without having to measure every single member of the population.

What is Estimation?

Estimation is the process of using sample data to approximate an unknown population parameter. By analyzing a sample, statisticians can draw conclusions and make predictions about the larger group from which the sample was drawn.

Types of Estimation

There are two primary types of estimation:

Point Estimation

Point estimation provides a single, best-guess value for an unknown population parameter. This single value is calculated directly from the sample data.

Example: The sample mean, denoted as $\bar{x}$, is commonly used as a point estimate for the population mean, denoted as $\mu$.

Interval Estimation

Interval estimation provides a range of plausible values, known as a confidence interval, within which the true population parameter is likely to lie. This approach acknowledges the inherent uncertainty in using sample data.

Example: A 95% confidence interval for the population mean might be reported as (15.2, 17.8). This means we are 95% confident that the true population mean falls within this range.

Why Estimation is Important?

Estimation plays a crucial role in statistical analysis and decision-making for several reasons:

• Inference about Populations: It allows us to draw conclusions about entire populations without the often prohibitive cost or impossibility of measuring every individual member.
• Decision-Making: Estimated values and their associated uncertainties inform critical decisions in various fields, including business, medicine, and policy.
• Forecasting: Estimation techniques are vital for predicting future trends and outcomes based on current data.
• Scientific Research: It enables researchers to test hypotheses and generalize findings from experimental samples to broader populations.
• Quantifying Uncertainty: Interval estimation (confidence intervals) provides a measure of the reliability of our estimates, helping us understand the potential range of error.

Common Estimators

The table below outlines common population parameters and their typical point estimators derived from sample data:

Key Concepts in Estimation

Several key concepts are important for understanding the quality and behavior of estimators:

Bias

Bias refers to the difference between an estimator's expected value and the true value of the population parameter it is estimating. An unbiased estimator has an expected value equal to the true parameter.

Consistency

An estimator is consistent if it gets closer and closer to the true population parameter as the sample size increases. Larger samples generally lead to more reliable estimates.

Efficiency

An estimator is considered efficient if it has the smallest possible variance among all unbiased estimators for a given parameter. An efficient estimator produces estimates that are more tightly clustered around the true parameter value.

Example of Estimation

A practical example of estimation involves using the average height of a sample of 100 people from a city to estimate the average height of the entire city's population. The calculated average height from these 100 individuals serves as a point estimate for the city's overall average height. A confidence interval could then be calculated to provide a range of plausible values for the city's true average height.