Study design

General Mathematics

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Units 3 & 40/70

Investigating data distributions0/6

types of data
16
representation, display and description of the distributions of categorical variables: data tables, two-way frequency tables and their associated segmented bar charts
6
representation, display and description of the distributions of numerical variables: dot plots, stem plots, histograms; the use of a logarithmic (base $10$ ) scale to display data ranging over several orders of magnitude and their interpretation in terms of powers of ten
14
use of the distribution(s) of one or more categorical or numerical variables to answer statistical questions
20
summary of the distributions of numerical variables; the five-number summary and boxplots (including the use of the lower fence ( $Q_1 - 1.5 \times \text{IQR}$ ) and upper fence ( $Q_3 + 1.5 \times \text{IQR}$ ) to identify and display possible outliers); the sample mean and standard deviation and their use in comparing data distributions in terms of centre and spread
69
the normal model for bell-shaped distributions and the use of the $68$ – $95$ – $99.7\%$ rule to estimate percentages and to give meaning to the standard deviation; standardised values ( $z$ -scores) and their use in comparing data values across distributions
31

Investigating association between two variables0/7

response and explanatory variables and their role in investigating associations between variables
contingency (two-way) frequency tables, their associated bar charts (including percentage segmented bar charts) and their use in identifying and describing associations between two categorical variables
12
back-to-back stem plots, parallel dot plots and boxplots and their use in identifying and describing associations between a numerical variable and a categorical variable
11
scatterplots and their use in identifying and qualitatively describing the association between two numerical variables in terms of direction (positive/negative), form (linear/non-linear) and strength (strong/moderate/weak)
7
answering statistical questions that require a knowledge of the associations between pairs of variables
Pearson correlation coefficient, $r$ , and its calculation and interpretation
8
cause and effect; the difference between observation and experimentation when collecting data and the need for experimentation to definitively determine cause and effect
4

Investigating and modelling linear associations0/8

least squares line of best fit $y = a + bx$ , where $x$ represents the explanatory variable, and $y$ represents the response variable; the determination of the coefficients $a$ and $b$ using technology, and the formulas $b = r \frac{s_y}{s_x}$ and $a = \bar{y} - b\bar{x}$
27
identification of the explanatory and response variables
8
use of the least squares method to fit a linear model to the data
6
use of the rule of the fitted line to make predictions being aware of the limitations of extrapolation
19
use of the coefficient of determination, $r^2$ , to assess the strength of the association in terms of explained variation
17
use of residual analysis to check quality of fit
23
data transformation and its use in transforming some forms of non-linear data to linearity using a square, logarithmic (base $10$ ) or reciprocal transformation (applied to one axis only)
4
interpretation and use of the equation of the least squares line fitted to the transformed data to make predictions
18

Investigating and modelling time series data0/5

qualitative features of time series plots; recognition of features such as trend (long-term direction), seasonality (systematic, calendar related movements) and irregular fluctuations (unsystematic, short-term fluctuations); possible outliers and their sources, including one-off real-world events, and signs of structural change such as a discontinuity in the time series
9
numerical smoothing of time series data using moving means with consideration of the number of terms required (using centring when appropriate) to help identify trends in time series plot with large fluctuations
14
graphical smoothing of time series plots using moving medians (involving an odd number of points only) to help identify long-term trends in time series with large fluctuations
9
seasonal adjustment including the use and interpretation of seasonal indices and their calculation using seasonal and yearly means
18
modelling trend by fitting a least squares line to a time series with time as the explanatory variable (data de-seasonalised where necessary), and the use of the model to make forecasts (with re-seasonalisation where necessary) including consideration of the possible limitations of fitting a linear model and the limitations of extending into the future
13

Depreciation of assets0/3

use of a first-order linear recurrence relation of the form $u_0 = a$ , $u_{n+1} = R u_n + d$ where $a$ , $R$ and $d$ are constants to generate the terms of a sequence
10
use of a recurrence relation to model and compare (numerically and graphically) flat rate, unit cost and reducing balance depreciation of the value of an asset with time, including the use of a recurrence relation to determine the depreciating value of an asset after $n$ depreciation periods for an initial sequence
26
use of the rules for the future value of an asset after $n$ depreciation periods for flat rate, unit cost and reducing balance depreciation and their application
23

Compound interest investments and loans0/4

the concepts of simple and compound interest
3
use of a recurrence relation to model and analyse (numerically and graphically) a compound interest investment or loan, including the use of a recurrence relation to determine the value of the compound interest loan or investment after $n$ compounding periods for an initial sequence from first principles
16
the difference between nominal and effective interest rates and the use of effective interest rates to compare investment returns and the cost of loans when interest is paid or charged, for example, daily, monthly, quarterly
9
the future value of a compound interest investment or loan after $n$ compounding periods and its use to solve practical problems
10

Reducing balance loans0/3

use of a first-order linear recurrence relation to model and analyse (numerically and graphically) the amortisation of a reducing balance loan, including the use of a recurrence relation to determine the value of the loan or investment after $n$ payments for an initial sequence from first principles
15
use of a table to investigate and analyse the amortisation of a reducing balance loan on a step-by-step basis, the payment made, the amount of interest paid, the reduction in the principal and the balance of the loan
14
use of technology with financial modelling functionality to solve problems involving reducing balance loans, such as repaying a personal loan or a mortgage, including the impact of a change in interest rate on repayment amount, time to repay the loan, total interest paid and the total cost of the loan
18

Annuities and perpetuities0/4

use of a first-order linear recurrence relation to model and analyse (numerically and graphically) the amortisation of an annuity, including the use of a recurrence relation to determine the value of the annuity after $n$ payments for an initial sequence from first principles
12
use of a table to investigate and analyse the amortisation of an annuity on a step-by-step basis, the payment made, the interest earned, the reduction in the principal and the balance of the annuity
5
use of technology to solve problems involving annuities including determining the amount to be invested in an annuity to provide a regular income paid, for example, monthly, quarterly
6
simple perpetuity as a special case of an annuity that lasts indefinitely
15

Compound interest investment with periodic and equal additions to the principal0/3

use of a first-order linear recurrence relation to model and analyse (numerically and graphically) annuity investment, including the use of a recurrence relation to determine the value of the investment after $n$ payments for an initial sequence from first principles
5
use of a table to investigate and analyse the growth of an annuity investment on a step-by-step basis after each payment is made, the payment made, the interest earned and the balance of the investment
2
use of technology with financial modelling functionality to solve problems involving annuity investments, including determining the future value of an investment after a number of compounding periods, the number of compounding periods for the investment to exceed a given value and the interest rate or payment amount needed for an investment to exceed a given value in a given time
6

Matrices and their applications0/5

matrix arithmetic: the order of a matrix, types of matrices (row, column, square, diagonal, symmetric, triangular, zero, binary and identity), the transpose of a matrix, and elementary matrix operations (sum, difference, multiplication of a scalar, product and power)
48
inverse of a matrix, its determinant, and the condition for a matrix to have an inverse
17
use of matrices to represent numerical information presented in tabular form, and the use of a rule for the $a_{ij}$ th element of a matrix to construct the matrix
18
binary and permutation matrices, and their properties and applications
14
communication and dominance matrices and their use in analysing communication systems and ranking players in round-robin tournaments
19

Transition matrices0/4

use of the matrix recurrence relation: $S_0$ = initial state matrix, $S_{n+1} = T S_n$ or $S_{n+1} = L S_n$ where $T$ is a transition matrix, $L$ is a Leslie matrix, and $S_n$ is a column state matrix, to generate a sequence of state matrices (assuming the next state only relies on the current state)
37
informal identification of the equilibrium state matrix in the case of regular transition matrices (no noticeable change from one state matrix to the next state matrix)
8
use of transition diagrams, their associated transition matrices and state matrices to model the transitions between states in discrete dynamical situations and their application to model and analyse practical situations such as the modelling and analysis of an insect population comprising eggs, juveniles and adults
18
use of the matrix recurrence relation $S_0$ = initial state matrix, $S_{n+1} = T S_n + B$ to extend modelling to populations that include culling and restocking
18

Graphs and networks0/2

the concepts, conventions and terminology of graphs including planar graphs and Euler's rule, and directed (digraphs) and networks
39
use of matrices to represent graphs, digraphs and networks and their application
8

Exploring and travelling problems0/3

the concepts, conventions and notations of walks, trails, paths, cycles and circuits
3
Eulerian trails and Eulerian circuits: the conditions for a graph to have a Eulerian trail or a Eulerian circuit, properties and applications
16
Hamiltonian paths and cycles: properties and applications
16

Trees and minimum connector problems0/3

trees and spanning trees
6
minimum spanning trees in a weighted connected graph and their determination by inspection or by Prim's algorithm
6
use of minimal spanning trees to solve minimal connector problems
3

Flow problems0/2

use of networks to model flow problems: capacity, sinks and sources
1
solution of small-scale network flow problems by inspection and the use of the 'maximum-flow minimum-cut' theorem to aid the solution of larger scale problems
23

Shortest path problems0/2

determination of the shortest path between two specified vertices in a graph, digraph or network by inspection
9
Dijkstra's algorithm and its use to determine the shortest path between a given vertex and each of the other vertices in a weighted graph or network
1

Matching problems0/2

use of a bipartite graph and its tabular or matrix form to represent a matching problem
3
determination of the optimum assignment(s) of people or machines to tasks by inspection or by use of the Hungarian algorithm for larger scale problems
11

Scheduling problems and critical path analysis0/4

construction of an activity network from a precedence table (or equivalent) including the use of dummy activities where necessary
10
use of forward and backward scanning to determine the earliest starting times (EST) and latest starting times (LST) for each activity
12
use of earliest starting times and latest starting times to identify the critical path in the network and determine the float times for non-critical activities
27
use of crashing to reduce the completion time of the project or task being modelled
12