Sampling Distributions: Concepts, Properties, and Applications

Chapter 6: Sampling Distributions

Introduction

This chapter introduces the concept of sampling distributions, a foundational topic in inferential statistics. Understanding sampling distributions is essential for making valid inferences about population parameters based on sample statistics.

Population Parameters and Sample Statistics

Definitions

• Parameter: A numerical descriptive measure of a population. It is calculated using all observations in the population, and its value is almost always unknown.

• Sample Statistic: A numerical descriptive measure of a sample. It is calculated using the observations in the sample.

Table: Population Parameters and Corresponding Sample Statistics

Population Parameter	Sample Statistic
Mean: μ	Sample mean: $\bar{x}$
Median: η	Sample median: M
Variance: $\sigma^2$	Sample variance: $s^2$
Standard deviation: $\sigma$	Sample standard deviation: s
Proportion: p	Sample proportion: $\hat{p}$

The Concept of a Sampling Distribution

Definition

• Sampling Distribution: The probability distribution of a sample statistic calculated from a sample of n measurements.

Sampling distributions describe the variability of a statistic from sample to sample and are central to statistical inference.

Comparing Estimators

• Sample mean ($\bar{x}$) and sample median (M) are both estimators of the population mean (μ).

• Depending on the sample, either $\bar{x}$ or M may be closer to μ.

Example: In one sample, $\bar{x}$ may be closer to μ than M, while in another sample, M may be closer.

Sampling Distribution Example

• For a sample of n = 25 temperature measurements, the sampling distribution of $\bar{x}$ is approximately normal, centered at the population mean (μ).

Estimating Population Variance

• Different statistics can be used to estimate the population variance ($\sigma^2$), each with its own sampling distribution.

Figure: Sampling distributions for two statistics (A and B) estimating $\sigma^2$ may differ in spread and center.

Table: Outcomes for n = 2 Coin Tosses

Outcome (Toss 1, Toss 2)	Probability
H (x = 1), H (x = 1)	1/4
H (x = 1), T (x = 0)	1/4
T (x = 0), H (x = 1)	1/4
T (x = 0), T (x = 0)	1/4

Properties of Sampling Distributions

Unbiasedness and Minimum Variance

• Estimator: A rule or formula that tells us how to use sample data to calculate a single number to estimate a population parameter.

• Unbiased Estimator: An estimator whose sampling distribution has a mean equal to the parameter it estimates. $E(\hat{\theta}) = \theta$

• Biased Estimator: An estimator whose sampling distribution mean does not equal the parameter. $E(\hat{\theta}) \neq \theta$

• Minimum Variance: Among unbiased estimators, the one with the smallest variance is preferred.

Example: The sample mean $\bar{x}$ is an unbiased estimator of the population mean μ.

The Sampling Distribution of $\bar{x}$ and the Central Limit Theorem

Properties of the Sampling Distribution of $\bar{x}$

• The mean of the sampling distribution of $\bar{x}$ equals the mean of the sampled population: $E(\bar{x}) = \mu$

• The standard deviation of the sampling distribution of $\bar{x}$ (standard error) is: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

Central Limit Theorem (CLT)

• If a random sample of n observations is selected from any population with mean μ and standard deviation σ, then for sufficiently large n, the sampling distribution of $\bar{x}$ will be approximately normal with mean μ and standard deviation $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$.

• The larger the sample size, the better the normal approximation.

Example: For n = 25, the sampling distribution of $\bar{x}$ is nearly normal, even if the population distribution is not normal.

Summary

• Population parameters are typically unknown and estimated using sample statistics.

• Sampling distributions describe the variability of sample statistics and are crucial for statistical inference.

• Unbiasedness and minimum variance are desirable properties for estimators.

• The Central Limit Theorem allows us to use normal probability models for sample means when sample sizes are large.