QMB3200- Simple Linear Regression and Formulas

The model (simple linear regression)

Relates one explanatory (independent) variable x to one response (dependent) variable y.

Fitted equation

ŷ = b₀ + b₁x, where b₀ is the intercept and b₁ is the estimated slope.

Slope

The predicted change in y for a one-unit increase in x. Association does not by itself establish causation.

R²

The proportion of the variation in y explained by the model (equal to r2 in simple regression with an intercept).

Relative frequency

f / n

Percent frequency

(f / n) × 100

Approximate class width

(largest value − smallest value) / number of classes

Mean

x̄ = Σxi / n μ = Σxi / N

Weighted mean

Σ(wixi) / Σwi

Geometric mean

(x₁·x₂·...·xn)^1/2

Range

largest − smallest

IQR

Q3 − Q1

IQR outlier limits

Lower: Q1 − 1.5(IQR) Upper: Q3 + 1.5(IQR)

Sample variance / SD

s² = Σ(xi − x̄)² / (n − 1) s = √s²

Population variance / SD

σ² = Σ(xi − μ)² / N σ = √σ²

z-score

z = (x − x̄) / s z = (x − μ) / σ Outlier: |z| > 3

Chebyshev

At least 1 − 1/z² within z standard deviations (z > 1)

Empirical rule

About 68% / 95% / 99.7% within 1 / 2 / 3 standard deviations

Correlation

r = sxy / (sxsy) ρ = σxy / (σxσy)

Covariance (random variables)

Cov(x,y) = [Var(x+y) − Var(x) − Var(y)] / 2

Multi-step counting

(n1)(n2)...(nk)

Combinations

C(N,n) = N! / (n!(N − n)!)

Permutations

P(N,n) = N! / (N − n)!

Addition law

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Conditional probability

P(A|B) = P(A ∩ B) / P(B)

Independence

P(A|B) = P(A) or P(B|A) = P(B)

Expected value

E(x) = Σ x·f(x)

Variance (discrete)

Var(x) = Σ (x − μ)²·f(x)

Binomial

f(x) = C(n,x)p^x(1 − p)^(n−x) E(x) = np Var(x) = np(1 − p)

Poisson

f(x) = μ^x e^(−μ) / x!

Exponential

f(x) = (1/μ)e^(−x/μ)

Standard error of the mean

σ / √n

z statistic (σ known)

z = (x̄ − μ₀) / (σ / √n)

t statistic (σ unknown)

t = (x̄ − μ₀) / (s / √n), df = n − 1

Regression line

ŷ = b₀ + b₁x

R²

R² = r² (simple regression with an intercept)