STA 126 – Exam 2 Review

Vocabulary practice flashcards for STA 126 Exam 2 review covering scatterplots, linear regression, contingency tables, probability rules, and counting principles.

Explanatory Variable

The predictor variable used to help explain or predict changes in another variable.

Response Variable

The outcome variable that a researcher seeks to explain or predict.

Correlation Coefficient ($r$)

A numerical statistic that measures the direction and strength of a linear relationship between two quantitative variables, satisfying $-1 \le r \le 1$.

Lurking Variable

An unincluded variable that may help explain an observed association between two variables, highlighting that correlation does not establish causation.

Residual ($e$)

The difference between an observed response value and its predicted value, calculated as $e = y - \hat{y}$.

Least-Squares Regression Line

The unique fitted line that minimizes the sum of squared residuals for a set of bivariate data.

Coefficient of Determination ($R^2$)

The proportion of total variation in the response variable explained by the least-squares regression line.

Residual Plot Diagnostic: Random Scatter

A residual plot exhibiting random scatter around zero with no clear pattern and roughly constant vertical spread, indicating an appropriate linear model.

Residual Plot Diagnostic: Nonlinear Relationship

A residual plot showing a distinct curved pattern around zero, providing evidence of a nonlinear relationship between the variables.

Residual Plot Diagnostic: Nonconstant Variability

A residual plot displaying a fan or megaphone pattern, indicating changing vertical spread (nonconstant variance) across the values.

Residual Plot Diagnostic: Unusual Observation

A residual plot showing an isolated point far from the horizontal band around zero, signaling a possible unusual or influential observation.

Contingency Table

A two-way table that cross-classifies and summarizes counts for two categorical variables.

Marginal Distribution

A frequency or relative-frequency distribution of a single categorical variable formed using the row or column table totals.

Conditional Distribution

A distribution that gives the relative frequencies of one categorical variable within a specified category of the other variable.

Association (Categorical Variables)

A relationship between two categorical variables occurring when the conditional distribution of one variable changes across categories of the other.

Simpson's Paradox

A phenomenon in which an observed association between two variables reverses or disappears when a third variable is introduced into the analysis.

Experiment

A repeatable process whose individual outcome is uncertain.

Sample Space ($S$)

The set of all possible outcomes of a probability experiment.

Event

A collection or subset of outcomes from the sample space of an experiment.

Simple Event

An event that consists of exactly one outcome.

Empirical Approach to Probability

A method of estimating probability using relative observed frequencies from repeated trials: $P(E) \approx \frac{\text{frequency of } E}{\text{number of trials}}$.

Classical Approach to Probability

A method of computing probability under the assumption of equally likely outcomes: $P(E) = \frac{N(E)}{N(S)}$.

Venn Diagram: Union ($E \cup F$)

The graphical representation of the event '$E$ or $F$' ($E \cup F$), where all regions belonging to $E$, $F$, or both are shaded.

Venn Diagram: Intersection ($E \cap F$)

The graphical representation of the event '$E$ and $F$' ($E \cap F$), where only the region shared by both $E$ and $F$ is shaded.

Venn Diagram: Complement ($E^c$)

The graphical representation of the event 'not $E$' ($E^c$), where all regions of the sample space outside $E$ are shaded.

Venn Diagram: Difference ($E \text{ and not } F$)

The graphical representation of the event '$E$ and not $F$', where the region of circle $E$ excluding the intersection with $F$ is shaded.

Disjoint Events

Events that cannot occur at the same time and share no common outcomes.

Independent Events

Events for which the occurrence of one event does not change the probability or knowledge of the other occurring.

Disjoint Addition Rule

The probability rule stating that if events $E$ and $F$ cannot occur simultaneously, $P(E \cup F) = P(E) + P(F)$.

General Addition Rule

The probability rule stating that for any two events $E$ and $F$, $P(E \cup F) = P(E) + P(F) - P(E \cap F)$.

Complement Rule

The probability rule stating that the probability of an event not occurring is $P(E^c) = 1 - P(E)$.

Independent Multiplication Rule

The probability rule stating that if events $E$ and $F$ are independent, $P(E \cap F) = P(E)P(F)$.

Multiplication Rule for Counting

A counting rule stating that if an operation consists of $k$ successive steps with $n_1, n_2, \dots, n_k$ choices, the total number of outcomes is $n_1 \times n_2 \times \dots \times n_k$.

Factorial ($n!$)

The product of all positive integers less than or equal to $n$ ($n! = n(n-1)(n-2)\dots 3 \cdot 2 \cdot 1$), with $0! = 1$ and $1! = 1$.