Statistics: Bivariate Data, Probability, and Discrete Distributions

Comprehensive practice flashcards covering linear and nonlinear regression, correlation coefficient, residuals, counting techniques, basic probability rules, contingency tables, conditional probability, discrete random variables, and the binomial distribution.

What is the least-squares criterion for determining the best-fitting regression line?

The least-squares criterion specifies that the best-fitting line is the one that minimizes the sum of the squared residuals: $\sum e^2 = \sum (y - \hat{y})^2$.

What formulas are used to calculate the slope ($b_1$) and $y$-intercept ($b_0$) of the least-squares regression line?

The slope is given by $b_1 = r \frac{s_y}{s_x}$, and the $y$-intercept is given by $b_0 = \bar{y} - b_1 \bar{x}$, where $r$ is the linear correlation coefficient, $s_x$ and $s_y$ are the sample standard deviations, and $\bar{x}$ and $\bar{y}$ are the sample means.

Why is the point of averages $(\bar{x}, \bar{y})$ significant in linear regression?

The least-squares regression line always passes through the point of averages $(\bar{x}, \bar{y})$. Consequently, predicting the outcome for an average explanatory value ($x = \bar{x}$) always yields the average response ($\hat{y} = \bar{y}$).

What is extrapolation in regression analysis, and why is it considered poor practice?

Extrapolation is using the regression equation to make predictions for values of the explanatory variable $x$ outside the range of observed sample data. It should be avoided because there is no assurance that the observed linear relationship holds beyond that range.

How is a residual ($e$) defined for an observed data point $(x, y)$?

A residual is the difference between the observed value of the response variable and the value predicted by the regression line: $e = y - \hat{y}$.

What does a residual plot indicate about the appropriateness of a linear regression model?

If the residual plot exhibits a random, balanced scatter around zero without any clear pattern or trend, a linear regression model is justified. If a curved or systematic pattern appears, a linear model is inappropriate.

What does a curved pattern in a residual plot indicate regarding model selection?

A curved pattern indicates that the underlying relationship between the variables is nonlinear, meaning the least-squares linear regression line is not suitable and a nonlinear model should be considered.

What is the distinction between a regression outlier and an influential point?

A regression outlier is a point far from the regression line. An influential point is an observation whose inclusion or removal substantially alters the position or slope of the regression line, typically occurring at an extreme $x$-value.

How did Alaska serve as an influential point in the U.S. farmland versus land area data?

Alaska has a vast total land area but very little farmland. Its inclusion dramatically flattened the slope of the regression line and changed the correlation coefficient from $r = 0.710$ down to $r = -0.119$, demonstrating that it is an influential observation.

How is the total deviation $(y - \bar{y})$ partitioned in regression analysis?

The total deviation $(y - \bar{y})$ is divided into the explained difference $(\hat{y} - \bar{y})$, which is accounted for by the regression line, and the unexplained difference $(y - \hat{y})$, which is the residual.

What is the coefficient of determination ($r^2$) and how is it interpreted?

The coefficient of determination, $r^2 = \frac{\text{Explained variation}}{\text{Total variation}}$, represents the proportion of total variation in the response variable $y$ that is explained by the least-squares regression line with predictor $x$.

What are the core mathematical properties of the linear correlation coefficient ($r$)?

(1) $-1 \le r \le 1$; (2) it is unitless; (3) switching the roles of $x$ and $y$ does not alter its value; (4) it measures only linear relationships; and (5) it is not resistant to outliers.

How does fitting a quadratic polynomial compare to a linear regression model when data shows slight curvature?

Fitting a quadratic polynomial increases the explanatory power (raising $R^2$) and reduces the residuals relative to the linear fit by accounting for the data's curvature.

What does the Law of Large Numbers state regarding probability experiments?

As a probability experiment is repeated a large number of times, the relative frequency (proportion of occurrences) of an event approaches its theoretical probability.

What probability threshold defines an event as unusual in elementary statistics?

By convention, any event whose probability is less than $0.05$ ($P < 0.05$) is considered unusual.

What is the range of possible probabilities, and what do values of $0$ and $1$ represent?

For any event $A$, $0 \le P(A) \le 1$. If $P(A) = 0$, $A$ is an impossible (null) event. If $P(A) = 1$, $A$ is a certain event.

How does the Empirical Method approximate the probability of an outcome?

It approximates probability by repeating an experiment many times and computing the proportion of trials in which the outcome occurs: $\frac{\text{Frequency of outcome}}{\text{Total number of trials}}$.

If an experiment has $n$ equally likely outcomes and $m$ result in a win, how are the odds of winning defined?

The odds of winning are given by $m : (n - m)$, read as '$m$ to $n - m$'.

What is the Rule of Complements in probability?

The complement of event $A$, denoted $A^c$ or 'not $A$', is the event that $A$ does not occur. Its probability is calculated as $P(A^c) = 1 - P(A)$.

What does it mean for two events $A$ and $B$ to be mutually exclusive (disjoint)?

Events $A$ and $B$ are mutually exclusive if they cannot occur simultaneously, meaning their intersection is empty ($A \cap B = \emptyset$) and $P(A \text{ and } B) = 0$.

What is the General Addition Rule for any two events $A$ and $B$?

$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$.

What is the Special Addition Rule for mutually exclusive events?

If events $A_1, A_2, \dots, A_n$ are mutually exclusive, then $P(A_1 \text{ or } A_2 \text{ or } \dots \text{ or } A_n) = P(A_1) + P(A_2) + \dots + P(A_n)$.

What is the inclusion-exclusion formula for the union of three events $A$, $B$, and $C$?

$P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C)$.

In a contingency table, how are joint, marginal, and conditional probabilities distinguished?

A joint probability is the intersection of categories from two different variables (cell count / grand total). A marginal probability concerns a single category of one variable (row or column total / grand total). A conditional probability restricts the sample space to a specific category (cell count / row or column subtotal).

In a table of $100$ students with $40$ Math majors where $28$ passed, what are the marginal $P(\text{Math})$, joint $P(\text{Math and Passed})$, and conditional $P(\text{Passed}|\text{Math})$ probabilities?

Marginal: $P(\text{Math}) = \frac{40}{100} = 0.40$. Joint: $P(\text{Math and Passed}) = \frac{28}{100} = 0.28$. Conditional: $P(\text{Passed}|\text{Math}) = \frac{28}{40} = 0.70$.

What is the formula for the conditional probability of event $B$ given event $A$?

$P(B|A) = \frac{P(A \text{ and } B)}{P(A)}$, provided $P(A) > 0$.

What is the General Multiplication Rule for two events $A$ and $B$?

$P(A \text{ and } B) = P(A) \cdot P(B|A) = P(B) \cdot P(A|B)$.

What are the three equivalent criteria used to determine if events $A$ and $B$ are independent?

Events $A$ and $B$ are independent if and only if at least one of the following holds: (1) $P(A|B) = P(A)$; (2) $P(B|A) = P(B)$; or (3) $P(A \text{ and } B) = P(A) \cdot P(B)$.

What is the 5% rule of thumb regarding sampling without replacement?

When sampling without replacement from a finite population, if the sample size is less than $5\%$ of the population size ($n < 0.05N$), the sampled selections may be treated as independent.

How is the probability that an event occurs 'at least once' calculated for $n$ independent trials with success probability $p$?

$P(\text{At least once}) = 1 - P(\text{None}) = 1 - (1 - p)^n$.

What does the Fundamental Principle of Counting (Multiplication Rule of Counting) state?

If an operation consists of $k$ sequential stages where stage 1 can be done in $n_1$ ways, stage 2 in $n_2$ ways, …, and stage $k$ in $n_k$ ways, the entire operation can be completed in $n_1 \times n_2 \times \dots \times n_k$ ways.

What is the difference between a permutation and a combination when selecting $k$ items from $n$ distinct items?

A permutation is an ordered selection without replacement ($nPk = \frac{n!}{(n - k)!}$), whereas a combination is an unordered selection without replacement ($nCk = \frac{n!}{k!(n - k)!}$).

What two conditions must every discrete probability distribution $P(x)$ satisfy?

(1) For each possible value $x$, $0 \le P(x) \le 1$; (2) The sum of all probabilities over all possible values must equal 1: $\sum P(x) = 1$.

How is the mean (expected value) of a discrete random variable $X$ computed?

$\mu_X = E(X) = \sum [x \cdot P(x)]$, which is the sum of the products of each possible outcome $x$ and its corresponding probability $P(x)$.

What are the defining and computing formulas for the variance of a discrete random variable?

Defining formula: $\sigma_X^2 = \sum (x - \mu)^2 P(x)$. Computing formula: $\sigma_X^2 = \left[ \sum x^2 P(x) \right] - \mu^2$.

What are the four required conditions for a random variable $X$ to follow a binomial distribution?

(1) The experiment consists of a fixed number ($n$) of trials; (2) Each trial has only two mutually exclusive outcomes: success or failure; (3) The probability of success ($p$) remains constant on each trial; (4) The trials are independent.

What is the formula for the probability of obtaining exactly $x$ successes in a binomial distribution with parameters $n$ and $p$?

$P(x) = \binom{n}{x} p^x (1 - p)^{n - x} = \frac{n!}{x!(n - x)!} p^x (1 - p)^{n - x}$, for $x = 0, 1, 2, \dots, n$.

What are the formulas for the mean, variance, and standard deviation of a binomial random variable?

Mean: $\mu = np$. Variance: $\sigma^2 = np(1 - p)$. Standard deviation: $\sigma = \sqrt{np(1 - p)}$.

What is the difference between binompdf and binomcdf on the TI-84 Plus calculator?

binompdf(n, p, x) computes the point probability of exactly $x$ successes ($P(X = x)$), while binomcdf(n, p, x) computes the cumulative probability of at most $x$ successes ($P(X \le x)$).

Why is the number of men taller than $6\,\text{ft}$ among five men sampled from one family not a binomial random variable?

The trials are not independent because members of the same family share genetic factors that influence height, violating the assumption of independent trials with a constant probability of success.