BUS105e Examination – January Semester 2016 Statistics BUS105e Copyright © 2016 SIM University Examination – January Semester 2016 Page 1 of 23 You must answer ALL the questions. (Total 100 marks) Question 1 The National Agricultural Statistics Service publishes a weekly report on crop progress in the United States. The report contains updated statistics on the states of the major crops in the United States. For Question 1, you will refer to Appendix A which contains a copy of the Crop Progress that was released on October 27, 2014. (a) Describe any four (4) types of crops that are reported in the current report. Describe any four (4) states of crop progress. (8 marks) (b) The diagram below is obtained by plotting the percentages of Corn Harvested (x-axis) with the corresponding percentages of Soyabeans Harvested (y-axis) for the week ending October 26, 2014, using the common states that occur in both tables. Name the type of statistical chart that is used in the above diagram. Describe what this chart tells you about the percentages of crops harvested in different states for the week ending October 26, 2014. (8 marks) BUS105e Copyright © 2016 SIM University Examination – January Semester 2016 Page 2 of 23 (c) The following diagrams show the states of Corn Mature and Sorghum Mature for 3 week-endings respectively: Corn Mature Sorghum Mature Name the type of statistical chart that is used in these two diagrams. What could be the useful information with respect to the states of Corn Mature and Sorghum Mature that you can derive from these two diagrams? (9 marks) BUS105e Copyright © 2016 SIM University Examination – January Semester 2016 Page 3 of 23 Question 2 (a) A survey is conducted to study the number and brands of mobile phones owned by university students in Singapore. The following table is a summary of the survey: Mobile Phones – Numbers and Brands Percentage 25.2% 1 Apple only 28.3% 1 Samsung only * 10.1% 1 Other only 5.4% 2 Apple only 4.5% 2 Samsung only 2.4% 2 Other only 12.0% 1 Apple and 1 Samsung only 8.5% 1 Samsung and 1 Other only 3.6% 1 Apple and 1 Other only 100% Total *Other refers to a brand other than Apple or Samsung. (b) (i) Identify the proportion of students who have at least one (1) Samsung mobile phone. (4 marks) (ii) Estimate the probability that a randomly chosen university student in Singapore has either at least one (1) Samsung mobile phone or at least one (1) other mobile phone. (4 marks) A survey is conducted on the likings of university students towards the movie Lord of the Rings. The survey is ranked on a scale from 1 to 10. Altogether, 40 students were interviewed. Summary statistics produced by Excel shows the following report: BUS105e Copyright © 2016 SIM University Examination – January Semester 2016 Page 4 of 23 Assume that likings for the movie are normally distributed with population mean 6 and standard deviation 3. (i) Use the Central Limit Theorem to estimate the probability that the sample mean is lesser than or equal to the mean as shown in the table above. (8 marks) (ii) Write down the value of y, where y is 1 standard deviation greater than the population mean. Estimate the probability that the liking of a randomly chosen university student is smaller than y. (9 marks) Question 3 (a) A study is undertaken to determine if the truck drivers that belong to a union receive equal wages as those that do not. The wages (in dollars) of 30 truck drivers that belong to a union and the wages of 30 truck drivers that do not belong to a union are sampled and analysed. With Alpha set at 0.05, the following report is produced by Excel: Execute an appropriate hypothesis test to determine the claim, assuming that the standard deviations of the two populations are equal. Show the detailed steps and the appropriate test statistic when conducting the test. (10 marks) BUS105e Copyright © 2016 SIM University Examination – January Semester 2016 Page 5 of 23 (b) Miss Kathy Ho is the CEO of a small company that manufactures candies for international sales. To keep standards high and maintain a perfect branding, Miss Ho embarks on a stringent manufacturing process in which candies produced are all circular in shape, with a diameter that does not deviate too much from a mean that can be controlled by the manufacturing machine. One day, Miss Ho takes a sample of 50 candies and measures their diameters. She quickly summarises the data with Excel as follows: Miss Ho decides to apply the method of confidence interval to analyse the data. Help Miss Ho by answering the following questions: (i) State the 99% confidence interval for the mean candy diameter. Interpret the confidence interval in a way that is useful for Miss Ho. (5 marks) (ii) State the reason for selecting your particular statistical distribution and the assumptions that you have made when constructing the confidence interval. (4 marks) (iii) List down the factors that may affect the location as well as the width of the confidence interval. Describe how varying these factors may change the location and width. (6 marks) BUS105e Copyright © 2016 SIM University Examination – January Semester 2016 Page 6 of 23 Question 4 A market research company wishes to analyse the prices of used cars using the method of multiple regression. The company identifies the age, the brand and the engine capacity of a car to be the major factors that determine its market price. The age of used cars is measured in years. The engine capacity of a car is expressed in terms of cubic centimetres (cc). The brand of the car is a ranked category with ranks 1 (worst) to 10 (best). Data for 15 used cars from the market are collected and analysed. The multiple regression test is performed using Excel. The regression results are as follows. Answer the following questions with reference to the regression results: (a) Describe the relationship between the variables Price (Y), Age (X1), Brand (X2) and Engine Capacity (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 Age, Brand and Engine Capacity affect Price. (4 marks) (b) State the coefficient of multiple determination and the adjusted coefficient of multiple determination. How can we interpret these coefficients and why are they different? (4 marks) (c) A used car dealer engages the market research company for consultation on how to price a 4.5-year old Porsche with a 2500 cc engine. Under the ranking system devised by the company, Porsche is ranked number 10. Estimate the price of the car by using the regression model from (a). (4 marks) BUS105e Copyright © 2016 SIM University Examination – January Semester 2016 Page 7 of 23 (d) Execute a hypothesis test at the 5% significance level to determine whether all regression coefficients are zero. Write down your steps carefully. (6 marks) (e) Multiple regression, like any statistical model, is valid as far as the assumptions made about the model hold. State the assumptions of a multiple regression model. Explain how you would test whether these assumptions are met in the case of used car price. (7 marks) ----- END OF PAPER ----- BUS105e Copyright © 2016 SIM University Examination – January Semester 2016 Page 8 of 23 Appendix A BUS105e Copyright © 2016 SIM University Examination – January Semester 2016 Page 9 of 23 BUS105e Copyright © 2016 SIM University Examination – January Semester 2016 Page 10 of 23 BUS105e Copyright © 2016 SIM University Examination – January Semester 2016 Page 11 of 23 BUS105e Copyright © 2016 SIM University Examination – January Semester 2016 Page 12 of 23 BUS105e Copyright © 2016 SIM University Examination – January Semester 2016 Page 13 of 23 BUS105e Copyright © 2016 SIM University Examination – January Semester 2016 Page 14 of 23 BUS105e Copyright © 2016 SIM University Examination – January Semester 2016 Page 15 of 23 BUS105e Copyright © 2016 SIM University Examination – January Semester 2016 Page 16 of 23 Source: Crop Progress Released October 27, 2014, by the National Agricultural Statistics Service (NASS), Agricultural Statistics Board, United States Department of Agriculture (USDA). BUS105e Copyright © 2016 SIM University Examination – January Semester 2016 Page 17 of 23 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 © 2016 SIM University Examination – January Semester 2016 Page 18 of 23 BUS105e Copyright © 2016 SIM University Examination – January Semester 2016 Page 19 of 23 Source: Basic Statistics for Business & Economics by Lind Douglas A., Marchal William G. & Wathen, Samuel A. (2012) 8th Edition, McGraw-Hill BUS105e Copyright © 2016 SIM University Examination – January Semester 2016 Page 20 of 23 Source: Basic Statistics for Business & Economics by Lind Douglas A., Marchal William G. & Wathen, Samuel A. (2012) 8th Edition, McGraw-Hill BUS105e Copyright © 2016 SIM University Examination – January Semester 2016 Page 21 of 23 Source: Basic Statistics for Business & Economics by Lind Douglas A., Marchal William G. & Wathen, Samuel A. (2012) 8th Edition, McGraw-Hill BUS105e Copyright © 2016 SIM University Examination – January Semester 2016 Page 22 of 23 Source: Basic Statistics for Business & Economics by Lind Douglas A., Marchal William G. & Wathen, Samuel A. (2012) 8th Edition, McGraw-Hill BUS105e Copyright © 2016 SIM University Examination – January Semester 2016 Page 23 of 23 BUS105e Examination – January Semester 2017 Statistics BUS105e Copyright © 2017 SIM University Examination – January Semester 2017 Page 1 of 14 You must answer ALL the questions. (Total 100 marks) Question 1 Refer to the following distribution of monthly commissions that sales personnel in a company earn: Table 1.1 Monthly commissions earned by sales personnel We can draw the two charts below: Chart 1.1 Monthly commissions earned by sales personnel Chart 1.2 Monthly commissions earned by sales personnel BUS105e Copyright © 2017 SIM University Examination – January Semester 2017 Page 2 of 14 (a) Name the charts above. What are the class intervals for the two charts? Explain which chart is better. (5 marks) (b) Describe the relative frequency of salespersons whose monthly commissions are between $1,600 and $1,800. (4 marks) (c) Describe the relative frequency of salespersons whose monthly commissions are at least $1,600. (5 marks) (d) What can be observed from Chart 1.1? How would you like to proceed with the analysis if you are given the raw data? Discuss. (5 marks) (e) Can unequal class intervals be used in a frequency distribution? Please explain. (6 marks) Question 2 A company is collecting data on the number of days its 125 employees were absent from work last month. The following probability distribution shows the likelihood that people were absent for 0, 1, 2, 3, 4, or 5 days last month. Table 2.1 Probability distribution of the number of days absent last month (a) Identify the average number of days an employee was absent last month. (4 marks) (b) Calculate the variance and standard deviation of the number of days absent. (6 marks) (c) Identify the probability that an employee had more than one day absent last month. (5 marks) BUS105e Copyright © 2017 SIM University Examination – January Semester 2017 Page 3 of 14 (d) What is mode? Find the mode of days an employee was absent last month. (4 marks) (e) Find the standard error of the mean. Interpret its meaning. (6 marks) Question 3 An investigation of the effectiveness of a training program to improve customer relationships included a pre-training and post-training customer survey. A high survey score indicates a good customer relationship. Seven customers were randomly selected and completed both surveys. The survey scores are listed below in Table 3.1. Table 3.1 Pre- and post- training survey scores The descriptive statistics for the difference of the pre-training survey and post-training survey scores are shown in Table 3.2. The difference is computed as “𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 – 𝑝𝑝𝑝𝑝𝑝𝑝-𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑟𝑟𝑒𝑒”. Table 3.2 Descriptive statistics for the survey score difference Survey Score Difference of Post - Pre Training Mean Standard Error Median Mode Standard Deviation Sample Variance Kurtosis Skewness Range Minimum Maximum Sum Count Confidence Level (95.0%) BUS105e Copyright © 2017 SIM University Examination – January Semester 2017 0.787 2.000 2.000 2.082 4.333 1.807 1.242 6.000 0.000 6.000 14.000 7.000 1.925 Page 4 of 14 Table 3.3 shows Excel results from the paired t-test for the difference of pre- and posttraining survey mean scores. Table 3.3 Paired t-test for the difference of two means (a) Identify and find the mean difference of the customer score before and after training. (4 marks) (b) State the 95% confidence interval for the mean difference of the survey scores before and after training, based on Table 3.2. Comment on the accuracy of the calculated confidence interval. (6 marks) (c) Select and execute an appropriate hypothesis test at the 5% significance level to examine whether the training program has improved customer relationships. Refer to the Excel report in Table 3.3. (15 marks) Question 4 A manager at a local bank analysed the relationship between monthly salary and three independent variables: - Service: length of service, number of months working in the bank Gender: 0 = female, 1 = male Job: job type, 0 = clerical job, 1 = technical job. BUS105e Copyright © 2017 SIM University Examination – January Semester 2017 Page 5 of 14 The following ANOVA table summarises the regression results: Table 4.1 ANOVA table (a) Describe the relationship between monthly salary (Y), service (X 1 ), gender (X 2 ) and job (X 3 ) by writing down the linear equation. Interpret the coefficient estimates. (5 marks) (b) Discuss which of the following is true, based on the hypothesis test for the individual regression coefficient. A. B. C. D. All the regression coefficients are not equal to zero "Job" is the only non-significant variable in the model The intercept is the only significant variable in the model "Service" is the only significant variable in the model (4 marks) (c) Execute a hypothesis test to determine whether all regression coefficients are zero at the 5% significance level. Write down your steps with details. (8 marks) (d) The multiple coefficient of determination can be calculated as (1004346.771/2465481.367) = 0.407 or 40.7 percent. State and interpret the multiple coefficient of determination. (3 marks) (e) Estimate the monthly salary of a male employee who has been working as a technical staff in the bank for 4 years and 6 months. (5 marks) ----- END OF PAPER ----BUS105e Copyright © 2017 SIM University Examination – January Semester 2017 Page 6 of 14 Appendix A: Formula Sheet Population Mean µ= ∑X N [1-1] Sample Mean X = ∑X n [1-2] 𝜇𝜇𝑋𝑋 = ∑ 𝑃𝑃𝑖𝑖 𝑋𝑋𝑖𝑖 [1-3] Range = Largest value – Smallest value [1-4] Mean of a discrete random variable Range Population variance ∑( X − µ ) 2 σ = N [1-5] Population standard deviation σ= ∑( X − µ ) 2 N [1-6] Sample variance ∑( X − X ) 2 s = n −1 [1-7] Sample standard deviation s= ∑( X − X ) 2 n −1 [1-8] Variance of a discrete random variable Standard deviation of a discrete random variable BUS105e Copyright © 2017 SIM University Examination – January Semester 2017 2 2 𝜎𝜎2𝑋𝑋 = ∑ 𝑃𝑃𝑖𝑖 (𝑋𝑋𝑖𝑖 − 𝜇𝜇𝑋𝑋 )2 𝜎𝜎𝑋𝑋 = �∑ 𝑃𝑃𝑖𝑖 (𝑋𝑋𝑖𝑖 − 𝜇𝜇𝑋𝑋 )2 [1-9] [1-10] Page 7 of 14 X −µ Standard normal value z= Standard error of mean σx = z-value, µ and σ z= known σ σ n [2-1] [3-1] X −µ σ [3-2] n Confidence interval for µ , with σ known X ±z σ n Confidence interval for µ , with σ unknown X ±t s Testing a mean, σ z= known [4-1] [4-2] n X −µ σ [5-1] n Testing a mean, σ t= unknown X −µ s [5-2] n p= Sample proportion Coefficient of determination BUS105e Copyright © 2017 SIM University Examination – January Semester 2017 𝑟𝑟 2 = 𝑆𝑆𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 X n =1− [6-1] 𝑆𝑆𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 [6-2] Page 8 of 14 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 © 2017 SIM University Examination – January Semester 2017 Page 9 of 14 BUS105e Copyright © 2017 SIM University Examination – January Semester 2017 Page 10 of 14 Source: Basic Statistics for Business & Economics by Lind Douglas A., Marchal William G. & Wathen, Samuel A. (2012) 8th Edition, McGraw-Hill BUS105e Copyright © 2017 SIM University Examination – January Semester 2017 Page 11 of 14 Source: Basic Statistics for Business & Economics by Lind Douglas A., Marchal William G. & Wathen, Samuel A. (2012) 8th Edition, McGraw-Hill BUS105e Copyright © 2017 SIM University Examination – January Semester 2017 Page 12 of 14 Source: Basic Statistics for Business & Economics by Lind Douglas A., Marchal William G. & Wathen, Samuel A. (2012) 8th Edition, McGraw-Hill BUS105e Copyright © 2017 SIM University Examination – January Semester 2017 Page 13 of 14 Source: Basic Statistics for Business & Economics by Lind Douglas A., Marchal William G. & Wathen, Samuel A. (2012) 8th Edition, McGraw-Hill BUS105e Copyright © 2017 SIM University Examination – January Semester 2017 Page 14 of 14 BUS105e Examination – July Semester 2015 Statistics BUS105e Copyright © 2015 SIM University Examination – July Semester 2015 Page 1 of 12 You must answer ALL the questions. (Total 100 marks) Question 1 JLM Leather Manufacturing Ltd. provides contract manufacturing services to many branded products. Since year 2000, JLM has been tracking the number of production stoppages each month. The data are provided in Table 1 below. Table 1. JLM Production Stoppages 2000 to 2014 Year Jan 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 Month Total (a) Feb 6 2 1 6 2 4 2 3 3 2 0 0 2 1 1 35 Mar 7 3 5 3 4 3 5 5 0 3 2 1 1 2 1 45 Apr 16 4 5 2 4 2 3 3 3 7 1 1 4 3 4 62 May 17 14 2 7 3 4 2 0 6 5 7 4 2 5 3 81 Jun 18 15 12 5 2 6 3 5 8 5 7 6 5 4 1 102 Jul 30 18 16 5 2 11 8 7 2 6 5 6 2 9 2 129 Aug 23 13 10 8 9 13 6 4 6 1 0 1 3 4 3 104 Sep 9 9 7 5 6 10 3 7 6 5 4 3 5 5 5 89 Oct 5 14 7 10 11 8 7 2 6 3 3 8 4 7 4 99 Nov 7 3 12 4 6 5 1 3 5 2 6 5 4 4 5 72 Dec 5 1 4 4 3 2 6 1 5 3 3 0 3 1 2 43 2 0 0 3 2 1 0 0 1 2 1 0 0 0 0 12 Compute the mean and range of the stoppage data of these 180 months. Use these values to summarise and describe the data on production stoppages. Interpret these measures of location and variation. Hint: Note that totals, highest and lowest values have been provided in Table 1. (3 marks) (b) Recommend at least two (2) types of charts that can describe these data graphically. Explain why you chose these types of charts. (3 marks) (c) Discuss the shape of the year total data distribution. Are production stoppages falling, increasing or consistent over the years? (3 marks) (d) Based on your answers above, interpret and explain how these statistical analyses can support management decision making. Hint: You can provide examples from your work experience. (3 marks) BUS105e Copyright © 2015 SIM University Examination – July Semester 2015 Page 2 of 12 Year Total 145 96 81 62 54 69 46 40 51 44 39 35 35 45 31 873 (e) The chart below (Figure 1) shows the total of production stoppages from 2000 to 2014. Year Total Year Figure 1. JLM Production Stoppages 2000 to 2014 Name the type of statistical chart and describe what the chart says about production stoppage over the years. Without further statistical analysis, predict, with explanation, if the number of production stoppage will fall, increase or remain the same in year 2015. (6 marks) (f) If you are given a free hand, how will you organise and present the data to more effectively describe these data? Explain why your method is better. (7 marks) Question 2 (a) Delicioso Pizza Pte Ltd sells 12-inch or 30 cm pizzas in outlets around the country. To predict future orders, a sample of 250 transactions were collected and analysed. The relative frequencies are listed in Table 2. Table 2. Delicioso Pizza Toppings Topping BBQMeatlovers Hawaiian SuperSupremo Supreme (i) % Sold 34% 34% 16% 16% Verify that the relative frequencies shown here can be interpreted as a legitimate assignment of probabilities for the 4 flavours offered. (3 marks) BUS105e Copyright © 2015 SIM University Examination – July Semester 2015 Page 3 of 12 (ii) Identify the probability that a randomly BBQMeatlovers or Hawaiian topping. chosen pizza has (3 marks) (iii) (b) Identify the probability that a randomly chosen pizza does not have a Supreme topping. (3 marks) Delicioso Pizza is concerned that the pizza diameter is not 30cm. The descriptive statistics on pizza diameter is provided in Table 3. For the following questions, you may need to use some formulae. Formulae and distribution tables are provided at the end of this examination paper – refer to Appendices A and B. Table 3. Delicioso Pizza Diameter. Pizza Diameter in cm Mean 28.31 Standard Error 0.08 28.77 Median Mode 29.40 Standard Deviation 1.28 Sample Variance 1.63 (1.14) Kurtosis Skewness (0.39) Range 5.55 Minimum 25.51 Maximum 31.06 Sum 7,077.05 Count 100.00 (i) Assuming that the sample statistics are good approximates of the population statistics and that the diameter of pizzas is distributed according to a normal distribution. State the value x where x is 2 standard deviations larger than the mean and estimate the probability that the diameter of a randomly chosen pizza is larger than x. (8 marks) (ii) Without assuming that the diameters of pizzas are distributed according to the normal distribution, estimate the probability that the mean diameter is larger than 31cm. The sample standard deviation is estimated to be 1.28. (4 marks) (iii) Explain why sampling and choosing appropriate sampling methods are important in statistical analysis. (4 marks) BUS105e Copyright © 2015 SIM University Examination – July Semester 2015 Page 4 of 12 Question 3 Delicioso Pizza Ltd’s sister company Delicioso Fine Dining Ltd. has restaurants in the city that are near major tourist attractions in Singapore. The general manager, Mr. William Teo, will like to open another restaurant in the city. He has to forecast revenues and is accountable to the company board. He decided to use past data from their oldest restaurant in the city located in the Orchard Road area. 50 days were randomly sampled from year 2014. The data and summary statistics are presented in Table 4. Table 4. Delicioso Fine Dining Orchard Revenue 2014- A sample of 50 days. 2010 1875 1905 1909 2172 1954 1945 1913 2192 1967 (a) (b) 2237 2240 2214 1896 2137 1999 2201 2169 2233 2172 2175 2090 2086 1865 2055 2212 1979 2211 1998 1951 2227 2106 1996 2208 2027 1976 2125 2225 2013 2226 1924 1938 2014 2208 1888 2194 1971 1910 2145 2147 Orchard Revnue S$ Mean Standard Error Median Mode Standard Deviation Sample Variance Kurtosis Skewness Range Minimum Maximum Sum Count Confidence Level(95.0%) 2,068.60 17.60 2,070.50 2,172.00 124.48 15,496.49 (1.55) (0.07) 375.00 1,865.00 2,240.00 103,430.00 50.00 35.38 Mr. Teo would like to use statistics to describe the mean and variations of revenue. (i) State the 95% confidence interval for the mean revenue. Interpret this confidence interval. (6 marks) (ii) Discuss the key assumption(s) you have made in your construction of the confidence interval. (3 marks) (iii) Discuss the factors that affect the width of the confidence interval. (6 marks) The sales manager, Mr. Michael Wee, advised Mr. Teo to also look at the Chinatown outlet revenue. Similar revenue data were collected for the Chinatown outlet. The mean, standard error and standard deviation were found to be 2125.28, 21.23 and 150.10 respectively. Mr. Wee conducted a t-test of independent samples with equal variance. The output is listed on Table 5. BUS105e Copyright © 2015 SIM University Examination – July Semester 2015 Page 5 of 12 Table 5. Delicioso Fine Dining Orchard and Chinatown outlets t-test of independent samples. Mean Variance Observations Pooled Variance Hypothesized Mean Difference df t Stat P(T
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