Distinguish between different types of numerical data and different data collection processes.
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- Terence McGee
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1 Level: Diploma in Business Learning Outcomes Distinguish between different types of numerical data and different data collection processes. Introduce the course by defining statistics and explaining the difference between descriptive statistics and statistical inference. Discuss the importance of statistics and quantitative analysis in business decision-making. Examine the main sources of data (e.g. government statistics, company annual reports and electronic databases) and give examples of different types of data (e.g. primary and secondary data, continuous and discrete data, quantitative and categorical data). Distinguish between the four main measurement scales (i.e. the nominal, ordinal, interval and ratio scales). Compare and contrast alternative methods of collecting primary data, including the use of probability and non-probability sampling methods, e.g. simple random sampling, systematic sampling, quota sampling and stratification. Explain the main features of surveys, discuss the advantages and disadvantages of interviews and questionnaires, and explain the concepts of sampling error and bias. Discuss the principles of questionnaire design and ask candidates to design and assess a simple questionnaire to investigate a simple business problem. Make sure that you emphasise the meaning and importance of random sampling, and explain clearly the difference between quantitative and categorical data, as these are important concepts in statistics.
2 Level: Diploma in Business Learning Outcomes Present data effectively and compute and interpret a range of summary statistics. Review the methods of presenting data in appropriate tables and charts, with particular emphasis on frequency distributions, cumulative frequency distributions, histograms and cumulative frequency curves. Review measures of location, including the arithmetic and geometric means, median, mode and percentiles. Show how these can be calculated for grouped and ungrouped data. Review measures of dispersion, with particular emphasis on the standard deviation. Show how the standard deviation can be calculated for grouped and ungrouped data. Show how the coefficient of variation and coefficient of skewness can be calculated and interpreted. Make sure that candidates are given plenty of practice at calculating the standard deviation using an appropriate calculator. In the examination, it is not necessary to compute the standard deviation by hand and a lot of time can be saved by using the statistical mode on a calculator.
3 Level: Diploma in Business Learning Outcomes Calculate and manipulate index numbers and understand their applications. Define index numbers and explain the difference between price and quantity indices. Discuss the use of index numbers in measuring changes in business and economic variables. Show how to change the base period of an index number series. Show how unweighted and weighted price and quantity index numbers can be calculated, including the simple aggregate index, the arithmetic and geometric means of price relatives, and the Laspeyres and Paasche weighted indices. Compare the advantages and disadvantages of the Laspeyres and Paasche indices and calculate Fisher s ideal index. Show how a consumer price index and an index of industrial production might be calculated for a country and explain how these index numbers are to be interpreted. Encourage candidates to practise lots of exercises which involve applying the Laspeyres and Paasche formulae. The incorrect evaluation of these formulae is a common mistake in the examinations.
4 Level: Diploma in Business Learning Outcomes 3.1, Explain the basic concepts of probability and apply concepts of probability to analyse business decision-making under conditions of uncertainty. Define probability and review the basic rules of probability: the addition rule for mutually exclusive and non-mutually exclusive events, and the multiplication rule for independent and non-independent events. Explain the concepts of conditional probability and expected monetary values (EMVs) and show how probability trees can be used as an aid to calculate probabilities. Construct decision trees and show how they can be used, together with the calculation of EMVs, as an aid to business decision-making under conditions of uncertainty. Discuss the limitations of EMV analysis in business decision-making. Make sure that candidates are given plenty of practice at constructing decision trees: they can be tricky. Often examination questions on this topic can be quite long and involved, so encourage candidates to read the questions very carefully.
5 Level: Diploma in Business Learning Outcomes Explain the binomial, Poisson and normal probability distributions and apply them to compute probabilities. Review the meaning of a probability distribution and distinguish between discrete and continuous probability distributions. Explain the conditions under which the binomial distribution may be applied and show how binomial probabilities can be calculated. Explain the conditions under which the Poisson distribution may be applied (including the Poisson approximation to the binomial distribution) and show how Poisson probabilities can be calculated. Discuss the use of the binomial and Poisson distributions in solving business problems. Explain the characteristics of the normal distribution and discuss the conditions under which it can be applied. Show how probabilities can be calculated by using the normal distribution tables to find areas under the standard normal curve. Candidates should be given lots of exercises to practise using the normal distribution tables, both for finding areas under the standard normal curve and for finding areas under any other normal curve.
6 Level: Diploma in Business Learning Outcomes Apply the normal and t distributions in estimation and hypothesis testing and conduct chi-squared tests. Explain and discuss the importance of sampling theory and the central limit theorem. Discuss how the central limit theorem widens the application of the normal distribution (and the related t distribution). Show how the normal distribution can be used to determine confidence intervals for means and proportions, where either the population standard deviation is known or the sample size is large. Explain that where the population standard deviation is unknown and the sample size is small, the appropriate t distribution should be used to construct confidence intervals. Show how to calculate the sample size required to estimate population values to within given limits. Candidates should be exposed to as many exercises as possible to illustrate the calculation of confidence intervals and the determination of the sample size required to achieve a particular error level at a given level of confidence.
7 Level: Diploma in Business Learning Outcome 5.3 Apply the normal and t distributions in hypothesis testing. Discuss the meaning of hypothesis testing and explain its importance to business decision-making. Clarify the terms null hypothesis, alternative hypothesis and the level of significance. Distinguish between one-tailed and two-tailed tests and Type I and Type II errors. Show how hypothesis tests of a single mean and single proportion are conducted using the normal or t distribution (as appropriate), with an emphasis on the interpretation of results. Show how hypothesis tests of a difference between means or proportions are conducted using the normal or t distribution (as appropriate), with an emphasis on the interpretation of results. Candidates should be advised to treat all statistical hypothesis tests like scientific experiments. First the null and alternative hypotheses should be specified and the level of significance selected. Then the test statistic should be computed and compared with the appropriate critical value or values. Finally, an appropriate conclusion should be drawn, in which the null hypothesis is either rejected or not rejected.
8 Level: Diploma in Business Learning Outcome 5.4 Conduct chi-squared tests in hypothesis testing. Distinguish between parametric and non-parametric methods and discuss the range of applications of non-parametric methods in business. Show how the chi-squared statistic can be used in a goodness-of-fit test, with an emphasis on the interpretation of results. Discuss the importance of chi-squared tests in the analysis of crosstabulations constructed from questionnaire responses. Show how to calculate the expected frequencies in each cell of a crosstabulation. Show how the chi-squared statistic can be used in a test of independence between two categorical variables, with an emphasis on the interpretation of results. This involves (a) stating appropriate null and alternative hypotheses, (b) deciding on a significance level and determining the critical value of chi-squared from the table of chi-squared critical values, (c) calculating the test statistic and (d) drawing an appropriate conclusion. Emphasise the importance of the chi-squared test in business research, particularly market research, and its wide application in the early stages of the analysis of questionnaire results.
9 Level: Diploma in Business Learning Outcomes Apply correlation and regression analysis to identify the strength and form of relationships between variables. Explain the difference between correlation and regression analysis. Plot different scatter diagrams to illustrate positive and negative linear correlation and different strengths of linear association. Distinguish between causal and non-causal relationships. Discuss the possibility of spurious correlation, particularly in the case of time-series data. Use numerical examples to calculate Pearson s coefficient of correlation and Spearman s coefficient of rank correlation and interpret the results. Explain the difference between simple and multiple regression and linear and non-linear regression. Discuss the importance of regression analysis in the investigation of causal relationships, forecasting and for measuring the impact of change on key variables in business. Apply the method of least squares to estimate the coefficients of a twovariable linear regression equation, and plot the regression line on a scatter diagram. Explain how the regression equation can be used for making predictions by interpolation and extrapolation. Discuss the likely accuracy of these predictions. Introduce the concept of multiple regression and the interpretation of regression output produced by spreadsheet or statistics software. Candidates should be able to interpret the estimated coefficients and their associated t statistics and the coefficient of determination (or R-squared). Make sure that candidates are given plenty of practice at calculating the correlation and regression coefficients using an appropriate calculator. In the examination, it is not necessary to compute these statistics by hand and a lot of time can be saved by using the statistical mode on a calculator.
10 Level: Diploma in Business Learning Outcomes Demonstrate how time-series analysis can be used in business forecasting. Explain the various components of a time series: trend, cyclical variation, seasonal variation and random variation. Distinguish between the additive and multiplicative models. Plot time-series data on a graph and identify the main components. Show how a trend can be estimated using the method of moving averages, with appropriate adjustments, and plot the trend on the time-series graph. Show how a trend can be estimated by using simple linear regression analysis with time as the independent variable, and plot the trend on the time-series graph. Apply the additive and multiplicative models to quantify the seasonal factors in a time series. Use the seasonal factors to compute a seasonallyadjusted series. Show how simple forecasts can be made by extending the trend in an appropriate way and then taking account of the estimated seasonal factors. Discuss the likely accuracy of forecasts based on a relatively short time series. Candidates should be encouraged to practise calculating moving averages and linear regression trends using their calculators. To compute these by hand in the examination can be very time-consuming.
11 Level: Diploma in Business Learning Outcomes Explain how mathematical relationships can be applied in the solution of economic problems. Review the meaning of functions, equations and graphs of functions. Using examples, show how to solve simple equations and two-variable simultaneous equations. Explain how to use linear equations to represent demand and supply schedules in a competitive market for a good, and to plot these equations on a demand and supply graph. Solve the demand and supply equations simultaneously to find the equilibrium market price and quantity. Analyse the effects on the equilibrium market price and quantity of shifts in the demand and supply functions. In particular, analyse the effects of the imposition of a sales tax. Explain some of the statistical problems of estimating demand and supply functions in practice and discuss the limitations of demand and supply analysis in economics. It is worth remembering that in the examination, a sales tax of, say, 10 per unit is always assumed to shift the supply curve vertically upwards by 10. The effect can be analysed, therefore, by adding 10 to the supply equation, so that P = a + bq becomes P = (a + 10) + bq, where P is the market price and Q is the quantity supplied.
12 Level: Diploma in Business Learning Outcomes Explain how mathematical relationships can be applied in the solution of business problems. Show how to represent cost, revenue and profit schedules using linear equations. Explain the principles of break-even analysis and outline the assumptions on which break-even analysis is based. Work through an example of a business facing linear revenue and cost functions and calculate the quantity at which zero profits will be made. Calculate the quantity required to achieve a given profit or loss, and calculate the profit or loss associated with a given quantity sold. Show how to calculate the effects on the break-even level of output of changes in costs and/or price. Show how break-even analysis can be represented graphically and how changes in the cost and revenue functions can be illustrated. Discuss the importance of break-even analysis in business and summarise the main limitations of simple break-even analysis. Typical examination questions on this topic assume that a firm has a given price and a cost function with both variable cost and fixed cost components. Starting with this information, encourage candidates to practise calculating the break-even output level and the effects of changes in the price, on one or both of the cost components.
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