Master Review

When describing a distribution, always comment on SOCS: Shape (symmetric, skewed left/right, bimodal), Outliers, Center (mean or median), and Spread (range, IQR, standard deviation). For skewed distributions, the median is the preferred measure of center; the IQR is the preferred measure of spread.

A z-score standardizes an observation by measuring how many standard deviations it lies from the mean. Positive z = above mean, negative z = below mean. The standard normal distribution N(0,1) allows probability lookup via tables or calculators.

The correlation coefficient r measures the strength and direction of a linear relationship (−1 ≤ r ≤ 1). Correlation does NOT imply causation. Lurking variables can create spurious correlations. Only controlled experiments can establish cause-and-effect.

The LSRL minimizes the sum of squared residuals. The line always passes through (x̄, ȳ). The slope b₁ tells you predicted change in ŷ per unit increase in x. R² (coefficient of determination) tells the proportion of variation in y explained by the linear model with x.

A simple random sample (SRS) gives every individual and group of that size equal chance of selection. Bias sources: voluntary response bias (self-selection overrepresents strong opinions), nonresponse bias, undercoverage. Increasing sample size reduces variability but does NOT reduce bias.

A well-designed experiment uses: Control (compare treatments to a baseline), Randomization (assign subjects at random to remove confounding), Replication (use many subjects to reduce chance variation). Blocking reduces variability due to known nuisance variables. Blinding eliminates placebo/experimenter effects.

Probability of any event A: 0 ≤ P(A) ≤ 1. Complement rule, addition rule for mutually exclusive events, and general addition rule. Independence: A and B independent if P(A∩B) = P(A)·P(B).

For a discrete random variable X with probability distribution, the mean (expected value) is the weighted average of outcomes. Variance is the weighted average of squared deviations from the mean. For independent RVs: means add, variances add (never standard deviations).

BINS conditions: Fixed number of trials n, Binary outcomes (success/failure), Independent trials, Same probability of success p each trial. X ~ B(n, p).

The sampling distribution of x̄ has mean μ and standard error σ/√n. By the Central Limit Theorem, for large enough n (n ≥ 30), x̄ is approximately normal regardless of the population shape. For proportions, p̂ is approximately normal if np ≥ 10 and n(1−p) ≥ 10.

A C% confidence interval means: if we repeated the sampling process many times, C% of the intervals constructed would capture the true parameter. It does NOT mean a C% probability the parameter is in this specific interval (the parameter is fixed, not random).

When σ is unknown, use the t-distribution with df = n − 1. Conditions: SRS (or random), Normal population or n ≥ 30 (CLT), Independent (10% condition). The p-value measures how likely the observed statistic is if H₀ is true.

For inference about a population proportion p, use the z-test when np₀ ≥ 10 and n(1−p₀) ≥ 10 (for tests) or np̂ ≥ 10 and n(1−p̂) ≥ 10 (for intervals). Use p₀ (hypothesized value) in the SE for tests; use p̂ for CI.

Type I error (α): Rejecting a true H₀ (false positive). Type II error (β): Failing to reject a false H₀ (false negative). Power = 1 − β = probability of correctly rejecting a false H₀. Increasing α, n, or effect size all increase Power.

Comparing two independent population means. Use two-sample t-test (do NOT assume equal variances unless stated). For paired data, take differences and use one-sample t on the differences.

Used for two-way tables to test whether two categorical variables are associated. Expected counts must be at least 5 in each cell. df = (rows − 1)(columns − 1). H₀: variables are independent.

Tests whether observed frequency data matches a claimed distribution. H₀: the population follows the specified distribution. df = k − 1 where k = number of categories. Same expected count condition (≥ 5) applies.

Test H₀: β₁ = 0 (no linear relationship). Uses t-test with df = n − 2. The standard error of the slope (SE_b₁) comes from computer output. A small p-value gives evidence of a statistically significant linear relationship.

The geometric distribution counts the number of trials until the first success. Conditions are the same as binomial (BINS) except the number of trials is not fixed — it's the variable of interest. P(X=k) = (1−p)^(k−1)·p.

Compare two independent population proportions. For the test, use the pooled p̂ (combined successes / combined n) in the SE formula. For the CI, use separate p̂₁ and p̂₂.