MATH104

Definition: Population

The entire collection of individuals, objects, or measurements about which a statistical inquiry seeks information.

Definition: Sample

A subset of a population selected to undergo measurement or observation in a study.

Definition: Statistic

A numerical characteristic or summary measure calculated directly from sample data (e.g., sample mean x-bar).

Definition: Parameter

A fixed (usually unknown) numerical characteristic of an entire population (e.g., population mean μ).

Definition: Sample Mean (x-bar)

The arithmetic average of the sample values: x-bar = (1/n) * ∑ x_i.

Definition: Sample Variance (s²)

The measure of data spread in a sample, computed using a denominator of (n - 1) to remain unbiased: s² = [1/(n-1)] * ∑ (x_i - x-bar)².

Theorem: Central Limit Theorem (CLT)

As sample size n becomes large (typically n ≥ 30), the sampling distribution of the sample mean becomes approximately normally distributed, regardless of the population's underlying distribution.

Definition: Point Estimator

A statistic (a rule or formula) used to estimate an unknown population parameter based on sample data.

Definition: Unbiased Estimator

An estimator whose expected value is exactly equal to the true population parameter it is trying to estimate: E[θ-hat] = θ.

Definition: Confidence Interval

An interval computed from sample data that contains the true population parameter with a specified probability (the confidence level).

Definition: Margin of Error

The maximum expected difference between the true population parameter and the sample point estimate at a given confidence level.

Definition: Null Hypothesis (H0)

A statement of no effect, no change, or no difference that is assumed true until evidence indicates otherwise.

Definition: Alternative Hypothesis (H1)

The statement that a statistical test sets out to find evidence for; it contradicts the null hypothesis.

Definition: Type I Error

The mistake of rejecting the null hypothesis (H0) when it is actually true (a false positive).

Definition: Type II Error

The mistake of failing to reject the null hypothesis (H0) when it is actually false (a false negative).

Definition: Significance Level (α)

The probability threshold for making a Type I error, typically set at 0.05.

Definition: P-value

The probability of obtaining sample results as extreme as, or more extreme than, the observed results, assuming the null hypothesis is true.

Definition: One-Sample t-Test

A hypothesis test used to compare the mean of a single sample to a known or hypothesized population mean when the population variance is unknown.

Definition: Paired t-Test

A statistical test used to compare the means of two dependent or related groups (e.g., before-and-after measurements).

Definition: Two-Independent-Samples t-Test

A statistical test used to determine if the means of two independent groups are significantly different from each other.

Definition: Linear Regression

A statistical method used to model the linear relationship between a dependent (response) variable and one or more independent (explanatory) variables.

Definition: Residual

The difference between an observed value and the value predicted by a regression model: e_i = y_i - y-hat_i.

Definition: Coefficient of Determination (R²)

A measure of the proportion of total variation in the dependent variable that is explained by the regression model.