Quantitative and analytical techniques for managers
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Section 1
- A researcher has collected a sample of data on the rate of return on assets of companies operating in the motor vehicles industry in Japan.
Table 1 Rate of Return on Assets of Japanese Motor Vehicle Manufacturers
Company |
Rate of Return on Assets (%) |
Company 1 | 6.1 |
Company 2 | 3.2 |
Company 3 | -5.0 |
Company 4 | 4.6 |
Company 5 | 11.5 |
Company 6 | 8.2 |
Company 7 | 21.3 |
Company 8 | -0.2 |
Company 9 | 10.1 |
Company 10 | 3.4 |
Company 11 | 5.3 |
Company 12 | 0.1 |
Company 13 | 8.4 |
Company 14 | 6.1 |
Company 15 | 5.9 |
Company 16 | 12.4 |
- Use the mean, mode, median, range, variance and standard deviation to describe the sample of data. Round all your answers to 1 decimal place.
- You have been asked by the Japanese Motor Vehicles Trade Association to write a 300-400 word report on the profitability of the companies and Motor Vehicles Manufacturers sector based on the above statistics. Provide your report including a discussion of the merits and dismerits of the statistical measures you have calculated in (a) and how these must be taken into consideration when interpreting the dataset in Table 1.
- You have collected data on the stock prices all companies listed on the Hong Kong Stock Market and have calculated that the mean share price is 510 and the standard deviation is 721. Round your answers to one decimal place and express your answers to (b) and (c) as percentages as well as probabilities.
- Explain what is meant by the z-score and the Standard Normal Distribution and show how they can be used to calculate probabilities.
- Calculate the probability that a company’s share price exceeds 600.
- Calculate the probability that a company’s share price is less than 250.
- You have been asked by a supermarket chain to test the hypotheses that advertising increases sales. You have been given time series data on sales and advertising for a leading manufacturing company. Answer ALL parts of this question:
Table 2 Advertising Expenditure and Sales, 2001-2010
Year |
Advertising Expenditure £m |
Sales £000m |
2001 | 50 | 32 |
2002 | 74 | 100 |
2003 | 19 | 12 |
2004 | 23 | 15 |
2005 | 82 | 61 |
2006 | 40 | 79 |
2007 | 186 | 125 |
2008 | 27 | 8 |
2009 | 20 | 12 |
2010 | 166 | 40 |
- Set up a Null Hypothesis and an Alternative Hypothesis to test the relationship between Advertising Expenditure and Sales.
- Use the data in Table 2 to calculate Pearson’s correlation coefficient between Advertising Expenditure and Sales. Give your answer to 1 decimal place.
- Calculate the t-ratio for Pearson’s correlation coefficient and determine whether the correlation coefficient is statistically significant at the 5% level using a one-tail test.
- In light of your findings what advice would you give to the company regarding the effectiveness of advertising?
Section 2
- A team of researchers aims to explore the hypothesis that productivity across countries is determined by investment in human capital. They have collected data on output per employee as a measure of productivity and the proportion of graduates in the workforce as a measure of human capital. The data are presented in Table 3. Using the cross sectional data in Table 3 answer all parts (a) to (e) below.
Table 3 Data on Productivity and Human capital for a sample of countries in 2005
Year (Time) |
Productivity ($000) P |
Human capital (% of workforce with Degree) H |
China | 26.1 | 11.2 |
Egypt | 24.3 | 15.3 |
Ghana | 19.6 | 12.4 |
India | 27.2 | 15.3 |
Indonesia | 15.6 | 10.5 |
Japan | 46.6 | 30.5 |
Singapore | 25.2 | 35.4 |
Vietnam | 20.1 | 13.6 |
UAE | 25.6 | 17.8 |
UK | 45.0 | 28.4 |
- Assuming that there is a linear relationship between productivity, P, and Human Capital, H, the researchers have set up the following linear model regression model:
P_{i} = β_{1} + β_{2}H_{i }+ e_{i}
Where the subscript i denotes country and e is a random error term. Calculate β_{1} and β_{2 }by hand using the Ordinary Least Squares (OLS) regression method.
- Explain the meaning of the constant term β_{1} and comment on its size and sign. Calculate the t-ratio for β_{1 }and determine whether β_{1 }significant at the 5% level. (Note that the standard error of the estimate of β_{1 }is 6.181).
- Explain the meaning of the slope coefficient β_{2} and comment on its size and sign. Carry out a t-test and state whether the coefficient is significant at the 5% level using a two-tail test. (Note that the standard error of the estimate of β_{2 }is 0.296).
- Explain the meaning of the R^{2 }statistic and the adjusted R^{2} The adjusted R^{2} statistic for this regression is 0.4. Comment on its size and provide an interpretation of this figure.
- e) Outline the limitations of the OLS model specified in (a) above and suggest how these might be addressed by further statistical work and diagnostic tests.
Section 3
Part three question and data to follow.
Write-up your results clearly and carefully. Please do not simply cut and paste output from SPSS – look at how the results are presented in the articles discussed in Tutorial 1 (on BLE) and follow that style of presentation. Note that results that reject hypotheses are just as valuable as results that confirm hypotheses. The key thing is to write up your results as clearly and impartially as possible using your knowledge of quantitative methods and regression analysis.
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