APPM1022A - Mathematical Statistics for Construction: Chapter 1 Sampling Distribution

Vocabulary practice cards defining key terms, sampling techniques, foundational formulas, and the Central Limit Theorem from Chapter 1 on Sampling Distributions.

Sampling Distribution

The probability distribution of a sample statistic (such as the sample mean or proportion), formed by calculating the statistic across repeated samples to show how much it varies around the true population parameter.

Sample

A set of observable random variables $X_1, X_2, \dots, X_n$, where the number $n$ is designated as the sample size.

Random Sample

A set of $n$ independent and identically distributed (iid) observable random variables $X_1, X_2, \dots, X_n$ drawn from a given population.

Identically Distributed

The condition where a set of random variables $X_1, X_2, \dots, X_n$ each possess the exact same probability distribution.

Statistic

A function $T$ of observable random variables $X_1, X_2, \dots, X_n$ that does not depend on any unknown parameters.

Central Limit Theorem (CLT)

A theorem establishing that for an independent and identically distributed random sample from a population with mean $\mu$ and finite variance $\sigma^2 > 0$, the standardized sample mean $\frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}}$ converges in distribution to a standard normal random variable $N(0, 1)$ as $n \rightarrow \infty$, regardless of the population distribution's shape.

Standard Error (SE)

The measure of spread or expected chance variation of a sample statistic; for a sample mean, it is defined mathematically as $SE = \frac{\sigma}{\sqrt{n}}$.

Stratified Sampling

The sampling method referred to as the 'Guaranteed Coverage' method, wherein every group is tested, but only a few individuals are drawn from each group.

Cluster Sampling

The sampling method known as the 'Energy-saver' method, wherein only a few groups are selected, but many individuals within those selected groups are tested.

Systematic Sampling

The sampling method known as the 'Assembly-Line' method, in which sample units are chosen according to a steady, predictable rhythm.

Standardized Sample Mean ($z$-score)

The measure of how many standard errors a sample mean is away from the expected population value, calculated as $z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$.

Population Proportion ($p$)

The true proportion of individuals within a given population who possess a particular characteristic or satisfy a specific binary condition.

Sample Proportion ($\hat{p}$)

The proportion calculated from a sample as $\hat{p} = \frac{x}{n}$, where $x$ represents the number of observed successes and $n$ is the sample size.

Standard Deviation of the Sample Proportion ($\sigma_p$)

The standard deviation of the sampling distribution of proportions, given by $\sigma_p = \sqrt{\frac{pq}{n}}$, where $p$ is the probability of success and $q = 1 - p$ is the probability of failure.

Standardized Sample Proportion ($z$-score)

The standardized test value for a sample proportion, calculated using the formula $z = \frac{\hat{p} - p}{\sqrt{\frac{pq}{n}}}$.

Sampling Distribution of Proportion Differences

The distribution formed by differences between two independent sample proportions ($\hat{p}_1 - \hat{p}_2$), which is closely normal when $n_1, n_2 \ge 30$, with mean $\mu_{(p_1 - p_2)} = p_1 - p_2$ and standard deviation $\sigma_{(p_1 - p_2)} = \sqrt{\frac{p_1 q_1}{n_1} + \frac{p_2 q_2}{n_2}}$.