BUS105e Examination – January Semester 2015 Statistics Thursday, 14 May 2015 10:00 am – 12:00 pm ____________________________________________________________________________________ Time allowed: 2 hours ____________________________________________________________________________________ INSTRUCTIONS TO STUDENTS: 1. This examination contains FOUR (4) questions and comprises TWENTYEIGHT (28) printed pages (including cover page, Appendices A and B). 2. You must answer ALL questions. 3. This is closed book examination. 4. All answers must be written in the answer book. At the end of the examination Please ensure that you have written your examination number on each answer book used. Failure to do so will mean that your work cannot be identified. If you have used more than one answer book, please tie them together with the string provided. THE UNIVERSITY RESERVES THE RIGHT NOT TO MARK YOUR SCRIPT IF YOU FAIL TO FOLLOW THESE INSTRUCTIONS. BUS105e Copyright © 2015 SIM University Examination – January Semester 2015 Page 1 of 28 You must answer ALL the questions. (Total 100 marks) Question 1 Demographic statistics are important for various aspects of social planning and organisation. For instance, the general fact that Singapore has low fertility rate - 1.19 in 2013 compared with a commonly recognised benchmark replacement rate of 2.1 features significantly in policy debates on immigration and economic development. For Question 1, you will refer to Appendix A which contains an abstract of The United Nations’ World Population Prospects: The 2012 Revision for data on global demographic estimates and projections. (a) Describe the population size and the male-to-female ratio of Singapore in 2013. Compare these statistics with those from the countries with the biggest (reported) and the smallest (reported) male-to-female ratios. (6 marks) (b) Discuss how data pertaining to population sizes and male-to-female ratios may be collected. How accurate do you think is the data? (6 marks) (c) From Table S.2 Total Population by Country, the following diagrams are produced with data selected from the years 2013 and 2025. BUS105e Copyright © 2015 SIM University Examination – January Semester 2015 Page 2 of 28 Name the type of statistical chart that is common to both diagrams. Explain why the bars in the top diagram appear like lines. Interpret what the charts say about the world population. (6 marks) (d) Countries accounting for about 75 per cent of the world population for the year 2013 (Table S.3) are shown here: BUS105e Copyright © 2015 SIM University Examination – January Semester 2015 Page 3 of 28 The following data table is produced by the Excel data analysis tools to summarise the statistics for these countries: Explain why the mean is so much bigger than the median and comment on the shape of the chart with reference to the world population situation using the data as shown in the table above. (7 marks) Question 2 (a) A survey is conducted with 100 Singaporeans on their favourite car brands by students in the Statistics 101 class. They tabulated the collected data as follows: Brand Mercedes Toyota BMW Mazda Others Percentage 17.5% 35% 13% 28% x% (i) Comment on the validity of the numerical figures shown under the Percentage column. Correct any error that you may detect. (ii) Taking into account your answer in (i), identify the value of x if the percentages shown are interpreted as the probabilities of favourite cars when a Singaporean is randomly asked. (iii) Identify the probability that a random favourite car response is BMW or Mercedes. (iv) Identify the probability that a random favourite car response is not categorised as “Others”. (9 marks) BUS105e Copyright © 2015 SIM University Examination – January Semester 2015 Page 4 of 28 (b) The IQ scores of 30 primary school students are: 89 89 89 93 93 98 98 98 98 102 102 105 105 105 107 110 110 110 114 114 119 119 119 119 123 123 126 126 126 128 On Excel, the summary statistics produced by the data analysis tools add-in looks like this: Assume that IQ scores for primary school students that have been kept for 50 years illustrate a normal distribution with a mean of 105 and a standard deviation that agrees with the above. (i) Write down the value of y where y is 1 standard deviation smaller than the population mean and estimate the probability that the IQ score of a randomly chosen primary school student is greater than y. (8 marks) (ii) Use the Central Limit Theorem to estimate the probability that the sample mean is greater than or equal to the mean as shown in the table above. (8 marks) BUS105e Copyright © 2015 SIM University Examination – January Semester 2015 Page 5 of 28 Question 3 Mr Jason Tan is the CEO of a company that produces cupcakes with a factory line production model. He has set in place stringent measures to ensure that the weights of the cupcakes do not deviate too much. To check that his measures work, he collected a random sample of 50 cupcakes and noted down their weights (in grams): Then Mr Tan applies the Excel data analysis tools to produce the descriptive statistics for the dataset which is shown as follows: (a) Mr Tan proceeded to use the method of confidence intervals to check the distribution of the weights of the cupcakes. (i) State the 99% confidence interval for the mean cupcake weight and interpret this confidence interval with respect to Mr Tan’s concern. BUS105e Copyright © 2015 SIM University Examination – January Semester 2015 Page 6 of 28 (ii) What is the difference if Mr Tan had used 95% instead of 99% for the construction of the confidence interval? (iii) If Mr Tan had thought that the stringent measures should produce cupcakes with a mean weight of 50 grams, what does the confidence interval say about this? (8 marks) Mr Tan isn’t so sure about his mean weight estimation for the cupcakes anymore. He relooked the manufacturing process measures and found some loopholes. He patched them up and redid the sampling and statistical experiment. The following set of data is obtained for the weights of another random sample of 50 cupcakes: (b) Mr Tan reapplied the Excel data analysis tools to produce the following descriptive statistics: BUS105e Copyright © 2015 SIM University Examination – January Semester 2015 Page 7 of 28 (c) (i) State the 99% confidence interval after Mr Tan patched up the loopholes. (ii) Compare the two (2) confidence intervals that have been obtained by Mr Tan in his statistical experiment and discuss what Mr Tan may conclude from them about his stringent measures to produce the target of 50 grams per cupcake consistently. (7 marks) For an alternative view, Mr Tan conducted a test of hypothesis on the second dataset of 50 cupcake weights. The following report is produced with his Excel application with the parameter Alpha set to 0.01: Select and execute an appropriate hypothesis test at the 0.01 significance level to help Mr Tan check whether the mean weight of 50 grams is acceptable. You are required to show the detailed steps and the appropriate test statistic in conducting the test. (10 marks) BUS105e Copyright © 2015 SIM University Examination – January Semester 2015 Page 8 of 28 Question 4 A property research company wishes to analyse the prices of luxury houses in Water Villa, a town of about 50 km2. For the purpose of future estimation, the company decides to use multiple regression to model house prices in terms of floor area, distance from town centre and age. Data from existing houses are collated and shown here: Excel is applied to perform the multiple regression test on the data. The following report is produced: BUS105e Copyright © 2015 SIM University Examination – January Semester 2015 Page 9 of 28 Answer the following questions with reference to the report: (a) Describe the relationship between the variables Price (Y), Floor Area (X1), Distance from Town Centre (X2) and Age (X3) by writing down the linear equation expressing Price as the dependent variable and the other factors as independent variables. Interpret the coefficients to show how Floor Area, Distance from Town Centre and Age impact Price. (4 marks) (b) State the coefficient of multiple determination and the adjusted coefficient of multiple determination. How may we interpret these coefficients and why are they different? (4 marks) (c) A new project is planned in which each unit of housing has a floor area of about 195 square metres. The distance of the plot of land under survey for the project is 7.5 km from the town centre. Estimate the price of each house that is to be built. (4 marks) (d) Conduct a hypothesis test to determine whether all regression coefficients are zero at the 5% significance level. Write down your steps carefully. (6 marks) (e) Multiple regression, like any statistical model, is valid as far as the assumptions made about the context hold. What are the assumptions of a multiple regression model and how do they fare under the present context of property prices? (7 marks) ----- END OF PAPER ----- BUS105e Copyright © 2015 SIM University Examination – January Semester 2015 Page 10 of 28 Appendix A BUS105e Copyright © 2015 SIM University Examination – January Semester 2015 Page 11 of 28 BUS105e Copyright © 2015 SIM University Examination – January Semester 2015 Page 12 of 28 BUS105e Copyright © 2015 SIM University Examination – January Semester 2015 Page 13 of 28 BUS105e Copyright © 2015 SIM University Examination – January Semester 2015 Page 14 of 28 BUS105e Copyright © 2015 SIM University Examination – January Semester 2015 Page 15 of 28 BUS105e Copyright © 2015 SIM University Examination – January Semester 2015 Page 16 of 28 BUS105e Copyright © 2015 SIM University Examination – January Semester 2015 Page 17 of 28 BUS105e Copyright © 2015 SIM University Examination – January Semester 2015 Page 18 of 28 BUS105e Copyright © 2015 SIM University Examination – January Semester 2015 Page 19 of 28 BUS105e Copyright © 2015 SIM University Examination – January Semester 2015 Page 20 of 28 BUS105e Copyright © 2015 SIM University Examination – January Semester 2015 Page 21 of 28 Source: United Nations Department of Economic and Social Affairs/Population Division World Population Prospects: The 2012 Revision, Highlights and Advance Tables BUS105e Copyright © 2015 SIM University Examination – January Semester 2015 Page 22 of 28 Appendix B Source: Basic Statistics for Business & Economics by Lind Douglas A., Marchal William G. & Wathen, Samuel A. (2012) 8th Edition, McGraw-Hill BUS105e Copyright © 2015 SIM University Examination – January Semester 2015 Page 23 of 28 BUS105e Copyright © 2015 SIM University Examination – January Semester 2015 Page 24 of 28 Source: Basic Statistics for Business & Economics by Lind Douglas A., Marchal William G. & Wathen, Samuel A. (2012) 8th Edition, McGraw-Hill BUS105e Copyright © 2015 SIM University Examination – January Semester 2015 Page 25 of 28 Source: Basic Statistics for Business & Economics by Lind Douglas A., Marchal William G. & Wathen, Samuel A. (2012) 8th Edition, McGraw-Hill BUS105e Copyright © 2015 SIM University Examination – January Semester 2015 Page 26 of 28 Source: Basic Statistics for Business & Economics by Lind Douglas A., Marchal William G. & Wathen, Samuel A. (2012) 8th Edition, McGraw-Hill BUS105e Copyright © 2015 SIM University Examination – January Semester 2015 Page 27 of 28 Source: Basic Statistics for Business & Economics by Lind Douglas A., Marchal William G. & Wathen, Samuel A. (2012) 8th Edition, McGraw-Hill BUS105e Copyright © 2015 SIM University Examination – January Semester 2015 Page 28 of 28
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