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SQA Higher Applications of Mathematics

Higher · SQAApplications of Mathematics38 notes in 6 folders, 235 KB

Notes for SQA Higher Applications of Mathematics (course specification version 2.0), in folders for the four content areas in the specification's order: Mathematical modelling; Statistics and probability; Finance; Planning and decision making. Each note has definitions, methods and worked examples, including the spreadsheet and statistical software skills the course uses. Delete any note you do not need once the notes are yours.

Adding them puts a copy in your notes, in a folder of its own with the folders below, for you to change and turn into flashcards or a question deck. Download gives you a zip of markdown files, which opens in any notes app.

What is inside

  • Mathematical modelling
    • Building a mathematical model6 KB
    • Linear, quadratic and exponential relationships6 KB
    • Recurrence relations4 KB
    • Choosing between models and improving them6 KB
    • Units and consistency5 KB
    • Error, tolerance and limits7 KB
    • Accuracy, precision and reading claims critically6 KB
    • Spreadsheets: presenting calculations clearly6 KB
    • Spreadsheets: functions, data and charts6 KB
  • Statistics and probability
    • Probability and data
      • Probability and independent events5 KB
      • Venn diagrams and tree diagrams6 KB
      • Types of data, populations and samples6 KB
      • Bias, outliers and data quality6 KB
      • Frequency tables, stem-and-leaf diagrams, bar and pie charts6 KB
      • Histograms and box plots6 KB
      • Contingency tables and misleading graphs6 KB
      • The shape of a distribution5 KB
      • Mean, standard deviation, median and interquartile range5 KB
    • Correlation, regression and testing
      • Scatter plots and correlation7 KB
      • Simple linear regression6 KB
      • Hypothesis testing6 KB
      • Confidence intervals6 KB
      • t-tests and paired t-tests6 KB
      • z-tests for two proportions5 KB
      • Errors, confounding variables and interpreting results7 KB
  • Finance
    • Effective rates of interest7 KB
    • Future and present value of a single payment6 KB
    • Series of payments7 KB
    • Changing, irregular and deferred payments7 KB
    • Loans and credit cards9 KB
    • Savings products7 KB
    • Insurance6 KB
    • Income and taxation6 KB
    • Inflation, purchasing power and financial strategies7 KB
  • Planning and decision making
    • Activity networks6 KB
    • Early and late times, float and the critical path7 KB
    • Gantt charts6 KB
    • Expected value and risk7 KB

The first note

Mathematical modelling / Building a mathematical model

A **mathematical model** is a description of a real situation in mathematical terms: variables, relationships between them, and the assumptions that make the relationships usable. It is always a simplification, so the useful question is whether it is good enough for the purpose, not whether it is exactly true. ## The modelling process Modelling moves round a loop rather than along a line. 1. Decide what the model is for, and state the question it has to answer. 2. Identify the quantities involved and choose the variables. 3. State the assumptions that simplify the situation. 4. Choose a form for the relationship and write it as a formula, a table, a graph or a chart. 5. Use the model to calculate or predict. 6. Compare the output with what happens in reality or with data, and judge whether it is plausible. 7. Change the assumptions or the form if the output is not good enough, and go round again. ## Variables and constants A **variable** is a quantity that changes. The **independent variable** is the one chosen freely or the one that drives the change, often time or distance, and the **dependent variable** is the one whose value is worked out from it. A quantity that stays fixed in the situation is a **constant** or **parameter**, such as a fixed booking fee or a growth rate that is assumed not to change. Suppose a taxi company charges a fixed £3.50 plus £1.80 for every mile. The independent variable is the distance $d$ in miles, the dependent variable is the fare $C$ in…

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