Stats Exam 2

Random variable

A variable whose possible outcomes are numeric values and whose probabilities are assigned according to some function or rule.

Random variable

A variable whose possible outcomes are numeric values and whose probabilities are assigned according to some function or rule.

Discrete random variable

A variable that takes on a finite or countably infinite number of values, often associated with counting (e.g., number of heads in coin flips).

Continuous random variable

A variable that can take on an infinite number of values within a given range, often associated with measurement (e.g., height, weight, time).

Probability model

The collection of all possible outcomes of a random variable and their corresponding probabilities. It can be organized in a table, formula, or graph.

Expected value (E[X])

The long-run average outcome from a probability model if the experiment is repeated indefinitely.

Standard deviation (σ or SD[X])

A measure of how much a random variable typically deviates from its expected value.

Uniform Model

A discrete probability distribution where all outcomes are equally likely. Formula: P(X = x) = 1/n. Parameters: Number of possible outcomes (n). Example: Rolling a fair die.

Binomial Model

A discrete probability distribution that calculates the probability of observing k successes in n independent trials. Formula: P(X = k) = (n choose k) p^k (1-p)^{n-k}. Parameters: n (trials), k (successes), p (probability of success). Expected value: E[X] = np. Standard deviation: SD[X] = sqrt(np(1-p)).

Normal Distribution

A continuous probability distribution that is symmetric and bell-shaped, defined by its mean (μ) and standard deviation (σ). Expected value: E[X] = μ. Standard deviation: SD[X] = σ.

Standard Normal Distribution

A normal distribution with mean μ = 0 and standard deviation σ = 1. Standardization formula: Z = (X - μ) / σ.

Sampling Distribution

The probability distribution of a sample statistic (e.g., sample mean) over many samples. Mean: E[X̄] = μ. Standard deviation: SD[X̄] = σ/√n.

Central Limit Theorem

The sampling distribution of the sample mean approaches a normal distribution as sample size increases, regardless of population shape.

Point Estimate

A single value used to estimate a population parameter. Example: Sample mean (X̄) estimates population mean (μ).

Confidence Interval

A range of values that likely contains the true population parameter. Formula: X̄ ± Z* (σ/√n). Confidence levels: 90% → Z* = 1.645, 95% → Z* = 1.960, 99% → Z* = 2.576.

Hypothesis Test

A procedure to determine if a sample provides enough evidence to reject a null hypothesis.

Null Hypothesis (H₀)

A statement that there is no effect or no difference (e.g., μ = μ₀).

Alternative Hypothesis (H₁ or Ha)

A statement that contradicts H₀ (e.g., μ ≠ μ₀, μ > μ₀, or μ < μ₀).

Test Statistic

Z = (X̄ - μ₀) / (σ/√n).

p-value

The probability of obtaining a sample result as extreme as the observed one, assuming H₀ is true.

Significance Level (α)

The threshold probability for rejecting H₀. Common values: α = 0.05 (5%), α = 0.01 (1%).

Type I Error

Rejecting H₀ when it is actually true.

Type II Error

Failing to reject H₀ when it is actually false.