QCAA General Maths EA flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Pearson correlation coefficient

A number between -1 and 1 describing the direction and strength of a linear association between two quantitative variables; sign gives direction, magnitude gives strength.

Coefficient of determination

The square of the correlation coefficient, r², interpreted as the proportion of variation in the response variable explained by the least-squares regression on the explanatory variable.

Least-squares regression line

The line y = a + bx minimising the sum of squared residuals; its slope equals r times sy divided by sx, and it passes through the mean point.

Residual

The vertical difference between an observed response value and the value predicted by the regression line, calculated as observed minus predicted; patterns in residuals signal model misfit.

Interpolation versus extrapolation

Interpolation predicts within the range of observed explanatory values and is generally reliable; extrapolation predicts outside that range and is unreliable because the pattern may not continue.

Lurking variable

An unmeasured variable associated with both the explanatory and response variables, which can create or distort an observed association and warn against causal conclusions.

Arithmetic sequence nth term

For first term a and common difference d, the nth term is tn = a + (n - 1)d; each term is obtained by adding the same constant to the previous term.

Arithmetic series sum

The sum of the first n terms of an arithmetic sequence is Sn = (n/2)(2a + (n - 1)d), equivalently n times the average of the first and last terms.

Geometric sequence nth term

For first term a and common ratio r, the nth term is tn = a·r^(n-1); consecutive terms are found by multiplying by the same nonzero constant.

Geometric series sum

For common ratio r not equal to 1, the sum of the first n terms is Sn = a(r^n - 1)/(r - 1); the formula applies to finite geometric series.

Limiting sum of geometric series

When the common ratio satisfies |r| < 1, the infinite geometric series converges to S∞ = a/(1 - r); otherwise the sum diverges and no limiting sum exists.

Great-circle distance

The shortest distance along Earth's surface between two points, calculated as the angular separation in degrees divided by 360, multiplied by 2πR with R about 6400 km.

Longitude and time difference

Earth rotates 360 degrees in 24 hours, so 15 degrees of longitude corresponds to one hour; time difference equals longitude difference divided by 15.

Small-circle distance

Distance travelled along a circle of latitude equals the angular difference in longitude divided by 360, multiplied by 2πR·cos(latitude).

Compound interest formula

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

Effective annual rate

The equivalent yearly growth rate under compounding, EAR = (1 + r)^n - 1, where r is the periodic rate and n is the number of compounding periods per year.

Appreciation and depreciation

Repeated percentage change modelled by A = P(1 ± r)^n; use plus for growth such as appreciation and minus for decay such as depreciation in value.

Reducing-balance loan recurrence

Balance after each payment follows An+1 = An(1 + r) - R, where r is the periodic interest rate and R the regular repayment; interest is charged on the remaining balance.

Annuity recurrence relation

The accumulated value of an annuity satisfies Vn+1 = Vn(1 + r) + R, where R is each regular deposit and r the interest rate per period.

Future value of an annuity

The total accumulated value of regular equal deposits earning compound interest, found by iterating the recurrence or summing the geometric series of individual deposits.

Present value

The single amount today equivalent to a future payment or stream of payments, obtained by discounting at the given interest rate over the relevant number of periods.

Perpetuity present value

For equal payments R continuing forever at periodic interest rate i, the present value is PV = R/i; the formula requires payments to be identical and evenly spaced.

Amortisation table

A period-by-period table listing opening balance, interest charged, payment, principal reduction and closing balance, where closing balance equals opening balance plus interest minus payment.

Vertex degree

The number of edges incident to a vertex in a network; summing all degrees gives twice the number of edges, since each edge contributes to two vertices.

Eulerian circuit

A closed trail traversing every edge of a connected graph exactly once; it exists only when every vertex of the graph has even degree.

Eulerian trail

An open trail using every edge exactly once; it exists in a connected graph only when exactly two vertices have odd degree, and those vertices are the endpoints.

Adjacency matrix

A square matrix whose entry in row i and column j counts the edges joining vertex i to vertex j; the sum of any row equals that vertex's degree.

Spanning tree

A connected subgraph containing every vertex of the original graph with no cycles; a spanning tree on n vertices always has exactly n - 1 edges.

Minimum spanning tree

A spanning tree of least total edge weight, found using Kruskal's algorithm by repeatedly adding the smallest-weight edge that does not create a cycle.

Critical path

The longest path through a project network from start to finish; activities on it have zero float and determine the minimum completion time.

Float time

The amount an activity can be delayed without delaying the whole project, equal to the latest finish time minus the earliest finish time for that activity.

Euler's planar graph formula

For a connected planar graph, the number of vertices minus the number of edges plus the number of faces equals 2, counting the outer region as a face.