- 0 (Registered)
Mathematics and its applications are quickly becoming indispensable in our modern technological world. Advancement in fields of applications has prompted the use of computational techniques unique to particular entities. The discipline of applied mathematics must respond to the demands of conceptual analysis, principles and problem solving for a new world filled with more advanced tools of technology. The main emphasis of the applied course is on developing the ability of the students to start with a problem in non-mathematical form and transform it into mathematical language. This will enable them to bring mathematical insights and skills in devising a solution, and then interpreting this solution in real-world terms. Students accomplish this by exploring problems using symbolic, graphical, numerical, physical and verbal techniques in the context of finite or discrete real-world situations.
This course aims to enable students to:
- equip themselves with tools of data collection, data organization and data analysis in order to make valid
decisions and predictions;
- develop research skills needed for productive employment, recreation and life-long education;
- use of appropriate statistical language and form in written and oral presentations;
- develop an awareness of the exciting applications of Mathematics;
- develop a willingness to apply Mathematics to relevant problems that are encountered in daily
- understand certain mathematical concepts and structures, their development and their
- use calculators and computers to enhance mathematical investigations;
- develop a spirit of mathematical curiosity and creativity;
- develop the skills of recognizing essential aspects of real-world problems and translating these problems
into mathematical forms;
- develop the skills of defining the limitations of the model and the solution;
- apply Mathematics across the subjects of the school curriculum;
- acquire relevant skills and knowledge for access to advanced courses in Mathematics and/or its
applications in other subject areas;
- gain experiences that will act as a motivating tool for the use of technology.
CAPE Applied Mathematics SBA
MODULE 1: COLLECTING AND DESCRIBING DATA (Source of Data) On completion of this Module, students should: 1. understand the concept of randomness and its role in sampling and data collection; 2. appreciate that data can be represented both graphically and numerically with the view to initiate analysis.
- Distinguish between qualitative and quantitative data, and discrete and continuous data;
- Distinguish between a population and a sample, a census and sample survey, and a parameter and a statistic;
- Identify an appropriate sampling frame for a given situation;
- Explain the role of randomness in statistical work;
- Explain why sampling is necessary;
- Outline the ideal characteristics of a sample;
- Distinguish between random and non-random sampling;
- Distinguish among the following sampling methods – simple random, stratified random, systematic random, cluster and quota;
- Use the ‘lottery’ technique or random numbers (from a table or calculator) to obtain a simple random sample;
- Outline the advantages and disadvantages of simple random, stratified random, systematic random, cluster and quota sampling
MODULE 1: COLLECTING AND DESCRIBING DATA (Data Collection)
- Design questionnaires, interviews and observation schedules;
- Use simple random, stratified random, systematic random, cluster and quota sampling to obtain a sample;
- Collect experimental data using questionnaires, interviews or observation schedules;
- Write a report of the findings obtained from collected data.
MODULE 1: COLLECTING AND DESCRIBING DATA (Data Analysis)
- Construct frequency distributions from raw data;
- Construct and use frequency polygons, pie charts, bar charts, histograms, stem-and-leaf diagrams, box-andwhisker plots and cumulative frequency curves (ogives);
- Outline the relative advantages and disadvantages of frequency polygons, pie charts, bar charts, histograms, stem-and-leaf diagrams and box-and-whisker plots in data analysis;
- Determine or calculate the mean, trimmed mean, median and mode for ungrouped and grouped data;
- Outline the relative advantages and disadvantages of the mean, trimmed mean, median and mode as measures of central tendency for raw or summarized data;
- Determine quartiles and other percentiles from raw data, grouped data, stem-and-leaf diagrams, box-andwhisker plots and cumulative frequency curves (ogives);
- Calculate the range, interquartile range, semi-interquartile range, variance and standard deviation of ungrouped and grouped data;
- Interpret the following measures of variability: range, interquartile range and standard deviation;
- Interpret the shape of a frequency distribution in terms of uniformity, symmetry, skewness, outliers and measures of central tendency and variability.
MODULE 2: MANAGING UNCERTAINTY (Probability Theory) On completion of this Module, students should: 1. understand the concept of probability; 2. appreciate that probability models can be used to describe real world situations and to manage uncertainty.
- List the elements of a possibility space (or probability sample space), given an experiment;
- Identify the elements of an event, given a possibility space;
- Calculate the probability of event A , P(A), as the number of outcomes of A divided by the total number of possible outcomes;
- Use the property that the probability of an event A is a real number between 0 and 1 inclusive (0 ≤ P(A) ≤ 1);
- Use the property that the sum of all the n probabilities of points
- Use the property that P(Á)= 1 –P(A), where P(Á) is the probability that event A does not occur;
- Calculate P(A ∪ B) and P(A ∩ B);
- Identify mutually exclusive events;
- Use the property of P(A ∩ B) = 0 or P(A ∪ B) = P(A) + P(B) where A and B are mutually exclusive events;
- Calculate the conditional probability where the probability that event A will occur given that event B has already occurred;
- Identify independent events;
- Use the property P(A ∩ B) = P (A) ⋅ P (B) or P(A|B) = P(A) where A and B are independent events;
- Construct and use possibility space diagrams, tree diagrams, Venn diagrams and contingency tables in the context of probability;
- Solve problems involving probability.
MODULE 2: MANAGING UNCERTAINTY (Random Variables)
- Use a given probability function which assigns probabilities to values of a discrete random variable;
- Outline and use the properties of the probability distribution of a random variable X:
- Calculate the expected value E(X), variance Var(X), and standard deviation of a discrete random variable X;
- Construct and use probability distribution tables for discrete random variables to obtain the probabilities: P(X = a), P(X > a), P(X < a), P(X ≥ a), P(X ≤ a), or any combination of these, where a and b are real numbers;
- Construct a cumulative distribution function table from a probability distribution table;
- Use a cumulative distribution function table to compute probabilities;
- Use the properties of a probability density function, f(x) of a continuous random variable X:
- Use areas under the graph of a probability density function as measures of probabilities (integration will not be tested), noting that P(X = a) = 0 for any continuous random variable X and real number a.
MODULE 2: MANAGING UNCERTAINTY (Binomial Distribution)
- State the assumptions made in modelling data by a binomial distribution;
- Identify and use the binomial distribution as a model of data, where appropriate;
- Use the notation X ~ Bin (n, p), where n is the number of independent trials and p is the probability of a successful outcome in each trial;
- Calculate and use the mean and variance of a binomial distribution;
- Calculate the probabilities P(X = a), P(X > a), P(X < a), P(X ≥ a) , P(X ≤ a), or any combination of these, where X ~ Bin (n, p).
MODULE 2: MANAGING UNCERTAINTY (Normal Distribution)
- Describe the main features of the normal distribution;
- Use the normal distribution as a model of data, as appropriate;
- Use the notation X ~ N (µ, σ² ), where µ is the population mean and σ² is the population variance;
- Determine probabilities from tabulated values of the standard normal distribution Z ~ N (0, 1);
- Solve problems involving probabilities of the normal distribution using z-scores;
- Explain the term ‘continuity correction’ in the context of a normal distribution approximation to a binomial distribution;
- Use the normal distribution as an approximation to the binomial distribution, where appropriate (np >5 and npq >5), and apply a continuity correction.
MODULE 3: ANALYSING AND INTERPRETING DATA (Sampling Distribution and Estimation) On completion of this Module, students should: 1. understand the uses of the sampling distribution and confidence intervals in providing information about a population; 2. understand the relevance of tests of hypotheses regarding statements about a population parameter; 3. appreciate that finding possible associations between variables and measuring their strengths are key ideas of statistical inference.
- Use the fact that E( x̄ ) = µ and Var ( x̄ ) = σ²/n where x̄ is the sample mean, µ the population mean, σ² the population variance and n the sample size;
- Apply the property that x̄ is normal if x is normal;
- Apply the Central Limit Theorem in situations where n ≥ 30;
- Calculate unbiased estimates for the population mean, proportion or variance;
- Explain the term ‘confidence interval’ in the context of a population mean or proportion;
- Calculate confidence intervals for a population mean or proportion using a large sample drawn from a population of known or unknown variance.
MODULE 3: ANALYSING AND INTERPRETING DATA (Hypothesis Testing)
- Formulate a null hypothesis H₀, and an alternative hypothesis H₁;
- Apply a one-tailed test or a two-tailed test, appropriately;
- Relate the level of significance to the probability of rejecting H₀ given that H₀ is true;
- Determine the critical values from tables for a given test and level of significance;
- Identify the critical or rejection region for a given test and level of significance;
- Evaluate from sample data the test statistic for testing a population mean or proportion;
- Apply a z-test
MODULE 3: ANALYSING AND INTERPRETING DATA (t-test)
- Evaluate the t-test statistic;
- Explain the term ‘degrees of freedom’ in the context of a t-test;
- Determine the appropriate number of degrees of freedom for a given data set;
- Determine probabilities from t-distribution tables;
- Apply a hypothesis test for a population mean using a small sample (n < 30) drawn from a normal population of unknown variance.
MODULE 3: ANALYSING AND INTERPRETING DATA (χ² -test)
- Evaluate the Chi-square test statistic
- Explain the term ‘degrees of freedom’ in the context of a χ² – test;
- Determine the appropriate number of degrees of freedom for a contingency table;
- Determine probabilities from χ² – tables;
- Apply a χ² test for independence in a contingency table (2×2 tables not included) where classes should be combined so that the expected frequency in each cell is at least 5.
MODULE 3: ANALYSING AND INTERPRETING DATA (Correlation and Linear Regression - Bivariate Data)
- Distinguish between dependent and independent variables;
- Draw scatter diagrams to represent bivariate data;
- Make deductions from scatter diagrams;
- Calculate and interpret the value of r, the product-moment correlation coefficient;
- Justify the use of the regression line y on x or x on y in a given situation;
- Calculate regression coefficients for the line y on x or x on y;
- Give a practical interpretation of the regression coefficients;
- Draw the regression line of y on x or x on y passing through x , y on a scatter diagram;
- Make estimations using the appropriate regression line;
- Outline the limitations of simple correlation and regression analyses.