Course Content

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 nonrandom 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, stemandleaf diagrams, boxandwhisker plots and cumulative frequency curves (ogives);
 Outline the relative advantages and disadvantages of frequency polygons, pie charts, bar charts, histograms, stemandleaf diagrams and boxandwhisker 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, stemandleaf diagrams, boxandwhisker plots and cumulative frequency curves (ogives);
 Calculate the range, interquartile range, semiinterquartile 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(AB) = 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 zscores;
 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 onetailed test or a twotailed 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 ztest

MODULE 3: ANALYSING AND INTERPRETING DATA (ttest)
 Evaluate the ttest statistic;
 Explain the term ‘degrees of freedom’ in the context of a ttest;
 Determine the appropriate number of degrees of freedom for a given data set;
 Determine probabilities from tdistribution 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 Chisquare 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 productmoment 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.