Key Concepts

Term: Probability Distribution
Definition: A function or table that describes all the possible values of a random variable and the probabilities associated with each value.

Term: Random Variable
Definition: A numerical variable that represents the outcome of a random phenomenon or experiment.

Term: Discrete Random Variables
Definition: Random variables that can take on a countable number of distinct values, such as whole numbers.

Term: Probability Distribution of a Discrete Random Variable
Definition: A list or function showing all possible values of a discrete random variable and the probability associated with each value.

Term: Probability Distribution Function (PDF)
Definition: A function that provides the probability that a discrete random variable equals a specific value.

Term: Cumulative Distribution Function (CDF)
Definition: A function that gives the probability that a random variable is less than or equal to a certain value.

Term: Mean and Standard Deviation of a Discrete Random Variable
Definition: The mean (expected value) is the average outcome weighted by probabilities, and the standard deviation measures the variability or spread of the possible values around the mean.

Term: Expected Value
Definition: The theoretical long-run average value of a random variable, calculated as the sum of all possible values weighted by their probabilities.

Term: Bernoulli Trials
Definition: Experiments or trials that have exactly two possible outcomes (success or failure) and a constant probability of success on each trial.

Term: Binomial Modeling
Definition: A statistical model that describes the number of successes in a fixed number of independent Bernoulli trials.

Term: Binomial Distributions
Definition: Probability distributions that summarize the likelihood of a given number of successes in a set number of Bernoulli trials.

Term: Continuous Random Variables
Definition: Random variables that can take on any value within a given interval or range, often associated with measurements.

Term: Probability Distribution of a Continuous Random Variable
Definition: Described by a probability density function (PDF), representing probabilities over intervals rather than specific values.

Term: Density Curves
Definition: Graphs that show the probability distribution of a continuous random variable; the area under the curve represents probabilities.

Term: Normal Distribution
Definition: A continuous probability distribution that is symmetric and bell-shaped, characterized by its mean (μ) and standard deviation (σ).

Term: Standardized Variable/Z-Score
Definition: A measure that describes a value's position relative to the mean in units of standard deviation; calculated as z=(X−μ)σz=σ(X−μ)​.

Term: Parameter
Definition: A numerical value that describes a characteristic of a population, such as the population mean (μ) or standard deviation (σ).

Term: Statistic
Definition: A numerical value calculated from sample data, used to estimate a population parameter, such as the sample mean (x̄).

Term: Sampling Error
Definition: The difference between a sample statistic and the corresponding population parameter, due to the randomness of sampling.

Term: Sampling Distribution
Definition: The probability distribution of a given statistic based on a random sample; shows how the statistic varies from sample to sample.

Term: Point Estimate
Definition: A single value calculated from sample data used to estimate a population parameter.

Term: Interval Estimate
Definition: A range of values derived from sample data that is likely to contain the population parameter with a certain level of confidence.

Term: Confidence Level
Definition: The probability that the interval estimate contains the population parameter; commonly used levels are 90%, 95%, and 99%.

Term: Margin of Error
Definition: The maximum expected difference between the true population parameter and a point estimate, accounting for sampling variability.

Term: Standard Error
Definition: The standard deviation of a sampling distribution; measures the variability of a statistic from sample to sample.

Term: T-Distribution
Definition: A probability distribution used when estimating population parameters when the sample size is small and/or population standard deviation is unknown.

Term: Rejection Region
Definition: The set of values for the test statistic that leads to rejecting the null hypothesis in a hypothesis test.

Term: Non-Rejection Region
Definition: The set of values for the test statistic where the null hypothesis is not rejected.

Term: Test Statistic
Definition: A standardized value calculated from sample data during a hypothesis test, used to decide whether to reject the null hypothesis.

Term: Significance Level (α)
Definition: The probability threshold set by the researcher (commonly 0.05) for rejecting the null hypothesis; represents the risk of a Type I error.

Term: P-Value
Definition: The probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true.

Term: Type I Error
Definition: The error made when the null hypothesis is true, but is incorrectly rejected.

Term: Type II Error
Definition: The error made when the null hypothesis is false, but is not rejected.