QCAA General Maths EA cheat sheet

Pearson correlation r measures strength of linear association: -1 ≤ r ≤ 1; r near 0 means weak linear association, not necessarily no association.

Coefficient of determination r² gives the proportion of variation in the response variable explained by the linear regression on the explanatory variable.

Least-squares regression line: y = a + bx, where b = r(sy/sx) and a = ȳ - b·x̄; the line always passes through (x̄, ȳ).

Residual = observed y - predicted y; a residual plot with no pattern supports a linear model, while curvature or fanning indicates the model is unsuitable.

Interpolation is prediction within the observed data range and is reliable; extrapolation outside that range is unreliable.

Correlation does not imply causation; a lurking variable may explain an association between two measured variables.

Arithmetic sequence: tn = a + (n-1)d; sum Sn = (n/2)(2a + (n-1)d) = (n/2)(a + tn).

Geometric sequence: tn = a·r^(n-1); sum Sn = a(r^n - 1)/(r - 1) for r ≠ 1, and S∞ = a/(1-r) only when |r| < 1.

Earth geometry: great-circle distance = (angular difference/360) × 2πR, and time difference = longitude difference ÷ 15° per hour, with R ≈ 6400 km.

Compound interest: A = P(1 + r)^n, where r is the interest rate per compounding period and n is the number of periods; divide annual rate by compounding frequency.

Effective annual rate compares compounding options: EAR = (1 + r)^n - 1, where r is the periodic rate and n periods occur per year.

Reducing-balance loan recurrence: An+1 = An(1 + r) - R, where An is the balance after n payments, r the periodic interest rate and R the payment.

Annuity future value recurrence: Vn+1 = Vn(1 + r) + R; the total of an annuity is the accumulated value of all deposits plus interest.

Perpetuity present value: PV = R/i, valid when payments continue indefinitely at periodic interest rate i.

Amortisation table columns: opening balance, interest charged, payment, principal reduction, closing balance; closing balance = opening + interest - payment.

In a network, the degree of a vertex is the number of edges incident to it; a graph has an Eulerian circuit only if it is connected and every vertex has even degree.

Euler's formula for a connected planar graph: v - e + f = 2; a spanning tree on n vertices has exactly n - 1 edges and no cycles.

Critical path analysis: earliest start time of an activity is the maximum earliest finish of its predecessors; the critical path has zero float and determines minimum project duration.