3- Sampling and Sampling Distributions

A comprehensive vocabulary review of sampling techniques, the sampling problem, sampling distributions, expected value, standard error, and the properties of the Central Limit Theorem.

Statistical Inference

The procedure of drawing conclusions or making inferences about population parameters (such as $\mu$, $\sigma^2$, and $\sigma$) based on sample statistics (such as $\bar{X}$, $s^2$, and $s$).

Simple Random Sampling

A sampling procedure in which every object or event drawn from the population always has an equal and consistent probability of being selected, ensuring statistical independence.

Opportunity Sample

A non-random sampling method where a researcher simply selects the first $N$ available units from a population that happen to come along or volunteer.

The Sampling Problem

The phenomenon where different random samples drawn from the same parent population yield different sample statistics (such as sample means), resulting in sample-to-sample variability.

Sampling Distribution

A theoretical distribution of a statistic obtained from infinite and repeated sampling of a specified sample size ($N$), with the values of the statistic plotted on the abscissa.

Sampling Distribution of the Mean

The theoretical probability distribution formed by calculating the means ($\bar{X}$) of an infinite number of random samples, each of the same size $N$, drawn from a population of scores.

Expected Value ($\mu_{\bar{X}}$)

The mean of a sampling distribution of the mean, which is equal to the mean of the parent population ($\mu$).

Unbiased Estimator

A sample statistic whose expected value over repeated infinite sampling equals the corresponding population parameter being estimated (e.g., $\mu_{\bar{X}} = \mu$).

Standard Error ($\sigma_{\bar{X}}$)

The standard deviation of a sampling distribution; for the mean, it is calculated as the population standard deviation divided by the square root of the sample size: $\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{N}}$.

Variance of the Sampling Distribution of the Mean ($\sigma_{\bar{X}}^2$)

The variance of the sampling distribution of the mean for a specified sample size, calculated as the parent population variance divided by the sample size: $\sigma_{\bar{X}}^2 = \frac{\sigma^2}{N}$.

Central Limit Theorem (CLT)

A theorem stating that for any population with mean $\mu$ and standard deviation $\sigma$, the sampling distribution of the mean will have mean $\mu_{\bar{X}} = \mu$, variance $\sigma_{\bar{X}}^2 = \frac{\sigma^2}{N}$, and will increasingly approximate a normal distribution as the sample size $N$ approaches infinity.

Normality Property of the Central Limit Theorem

The principle that as sample size increases (e.g., from $n = 5$ to $n = 30$), the sampling distribution of $\bar{X}$ converges toward a normal bell shape, regardless of whether the original population distribution is uniform, bimodal, or skewed.

Precision of an Estimate

The closeness of repeated sample estimates to one another, which increases as sample size ($N$) increases and standard error ($\sigma_{\bar{X}}$) decreases.