Statistics Exam 2 Review (Chapters 4-8)

Comprehensive vocabulary flashcards covering key terms, probability rules, normal distributions, sampling distributions, the Central Limit Theorem, and confidence intervals from Chapters 4 through 8.

Experiment

A process that yields an outcome, unknown until observed.

Sample Space ($S$)

The set of all possible outcomes of an experiment.

Event

A subset of outcomes; a simple event has exactly one outcome.

Mutually Exclusive Events

Events that cannot occur together (they have no overlap).

Exhaustive Events

Events that collectively cover the entire sample space.

Union ($A \cup B$)

The event containing outcomes that are in $A$ or $B$ (or both).

Intersection ($A \cap B$)

The event containing outcomes that are common to both $A$ and $B$.

Complement ($A^c$)

The event consisting of all outcomes in the sample space $S$ that are not in $A$.

Subjective Probability

Probability based on personal judgment or belief.

Empirical Probability

Probability calculated as the relative frequency from observed data.

Classical Probability

Probability computed as favorable outcomes divided by total outcomes when all outcomes are equally likely.

Conditional Probability

The probability that event $A$ occurs given that event $B$ has already occurred, denoted $P(A \mid B)$.

Independence

A condition in which the occurrence of one event does not change the probability of the other event.

Contingency Table

A frequency table for two categorical variables that supports calculating joint, marginal, and conditional probabilities.

Total Probability Rule

A method to decompose the probability of an event $P(A)$ across a partition (e.g., $B$ versus $B^c$): $P(A) = P(A \mid B)P(B) + P(A \mid B^c)P(B^c)$.

Bayes' Theorem

A rule used to update the prior probability $P(B)$ to the posterior probability $P(B \mid A)$ using likelihoods.

Normal Distribution $N(\mu, \sigma)$

A bell-shaped, symmetric probability distribution where $\text{mean} = \text{median} = \text{mode}$.

Standard Normal Distribution ($Z \sim N(0, 1)$)

A specific normal distribution with a mean of $\mu = 0$ and a standard deviation of $\sigma = 1$.

z-score

A standardized value representing the number of standard deviations a data point lies from the mean, calculated as $Z = \frac{X - \mu}{\sigma}$.

Cumulative Distribution Function ($\Phi$)

A function that gives the cumulative area under the standard normal curve to the left of a given value.

Quantile / Percentile

A value $x$ associated with a given cumulative probability $p$.

Empirical Rule

A rule for normal distributions stating that approximately $68\%$ of observations fall within $1\sigma$, $95\%$ fall within $2\sigma$, and $99.7\%$ fall within $3\sigma$ of the mean.

Population Parameter

A fixed (often unknown) numerical value that characterizes a population (e.g., $\mu$, $\sigma^2$).

Statistic

A numerical value computed from a sample (e.g., $\bar{X}$) that varies from sample to sample.

Sampling Distribution

The probability distribution of a statistic obtained from all possible samples of size $n$.

Estimator vs. Estimate

An estimator is the sample statistic used to estimate a population parameter, whereas an estimate is its realized numerical value.

Standard Error (SE)

The standard deviation of a sampling distribution; for the sample mean, it is $SE(\bar{X}) = \frac{\sigma}{\sqrt{n}}$.

Central Limit Theorem (CLT)

A theorem stating that for a large sample size $n$, the sample mean $\bar{X}$ is approximately normal regardless of the population shape under mild conditions.

t-Distribution

A bell-shaped distribution with heavier tails than the standard normal distribution, characterized by degrees of freedom ($df = n - 1$), that approaches the normal distribution as $df$ increases.

Degrees of Freedom ($df$)

The number of independent values that can vary in an analysis; for a one-sample mean with unknown $\sigma$, $df = n - 1$.

Confidence Interval (CI)

An interval computed from sample data used to estimate an unknown population parameter.

Confidence Level

The long-run capture rate of the confidence interval procedure (e.g., $90\%$, $95\%$, or $99\%$).

Margin of Error (ME)

The half-width of a confidence interval, representing the maximum expected difference between the true parameter and the sample estimate.