BUS105e Examination – July Semester 2015 Statistics Thursday, 12 November 2015 10:00 am – 12:00 pm ____________________________________________________________________________________ Time allowed: 2 hours ____________________________________________________________________________________ INSTRUCTIONS TO STUDENTS: 1. This examination contains FOUR (4) questions and comprises TWELVE (12) 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 – 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 Median 28.77 Mode 29.40 Standard Deviation 1.28 Sample Variance 1.63 Kurtosis (1.14) 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<=t) one‐tail t Critical one‐tail P(T<=t) two‐tail t Critical two‐tail  Orchard  Revenue S$              2,068.60           15,496.49                   50.00           19,012.61                       ‐                   98.00                    (2.06)                     0.02                     1.66                     0.04                     1.98  Chinatown  Revenue S$            2,125.28         22,528.74                 50.00 At 5% level of significance, select and execute a hypothesis testing and determine if the means of the revenues are different. All the relevant statistics are provided in Table 5. You are required to show the detailed steps. If you had a free hand, explain how you would like to conduct an appropriate hypothesis test and explain why your method is better. You are not required to work out the statistics. (10 marks) Question 4 Peligroso Security Ltd. provides security equipment and services to Delicioso outlets and restaurants. The general manager, Mr. Rueben Tan, plans to expand and offer security equipment and services to residential clients. Mr. Tan found that the security expenditure of residential clients is affected by area size, age of property, number of home appliances and type of property. He engaged some part-time staff to administer questionnaires at various Delicioso locations. A total of 1200 questionnaires were administered but only 1000 were found to be good enough for analysis. The data header descriptors are provided below: SQFT = Square feet of living space AGE = Age of property (years) APP = Number of home appliances (dishwasher, refrigerator, microwave, disposer, washer, intercom, skylight(s), compactor, dryer, handicap fit, cable TV access TYPE= Type of property. 1=HDB with 2 bedrooms or less, 2 = HDB with 3 bedroom, 3 = HDB with 4 or more bedrooms or Condominium with 2 bedroom or less, 4 = Condominium with 3 or more bedrooms and 5 = Landed Property BUS105e Copyright © 2015 SIM University Examination – July Semester 2015 Page 6 of 12 SEC_EXP = Annual Security Services Expenditure which includes equipment purchase and service expenditure in a year. Typically, condominium apartments are considered to offer better facilities than HDB flats and most residents will prefer more rooms for a similar location and price. Multiple regression analysis is applied in Excel to produce the following report: Table 6. Peligroso Security Regression Statistics SUMMARY OUTPUT Regression Statistics Multiple R 0.804089164 R Square 0.646559383 Adjusted R Square 0.645138516 Standard Error 284.1075783 Observations 1000 ANOVA df Regression Residual Total Intercept SQFT AGE APP TYPE 4 995 999 SS 146919918.8 80313530.46 227233449.3 MS F Significance F 36729979.7 455.0457388 6.2544E‐223 80717.11604 Coefficients Standard Error t Stat P‐value ‐55.52376697 38.5523902 ‐1.440215942 0.150120878 0.424884887 0.010022243 42.39419102 3.5622E‐225 ‐5.38654978 1.239224191 ‐4.346711289 1.52393E‐05 1.174483501 3.057445776 0.384138784 0.700957664 1.621754832 6.477768236 0.250357032 0.802362938 Lower 95% Upper 95% Lower 95.0% Upper 95.0% ‐131.1770879 20.12955394 ‐131.1770879 20.12955394 0.405217728 0.444552045 0.405217728 0.444552045 ‐7.818342592 ‐2.954756968 ‐7.818342592 ‐2.954756968 ‐4.82529823 7.174265233 ‐4.82529823 7.174265233 ‐11.08990005 14.33340971 ‐11.08990005 14.33340971 Reading from the report, answer the following questions: (a) Identify and fit a linear regression of the data by writing down the linear equation that relates the security expenditure (SEC_EXP) to area size (SQFT), age of property (AGE), number of appliances (APP) and type of property (TYPE). (3 marks) (b) State the coefficient of determination and the adjusted coefficient of determination. Interpret them and explain how they are different from each other. (3 marks) (c) Mr. Tan will like to open one store in a relatively new housing estate. On average, the size is approximated to be 1150 sq ft, age to be 5 years, each property has around 6 electrical appliances and 90% of the property are HDB 3 room flats. How much can Mr. Tan expect each household to spend per year on security in this estate? (3 marks) BUS105e Copyright © 2015 SIM University Examination – July Semester 2015 Page 7 of 12 (d) Execute a global hypothesis test on the regression model at the 5% significance level to investigate whether or not all the independent variables have zero regression coefficients. List down your steps carefully. (3 marks) (e) Execute a hypothesis test on each of the coefficients and interpret. Evaluate and decide which variables should be retained in your linear relationship and explain why. (6 marks) (f) What are the assumptions made when the method of regression is applied to study data? Discuss how violations of these assumptions will impact interpretation and conclusion. (7 marks) ----- END OF PAPER ----- BUS105e Copyright © 2015 SIM University Examination – July Semester 2015 Page 8 of 12 Appendix A: Formula Sheet Population Mean  X N [1-1] Sample Mean X  X n [1-2] Range Range = Largest value – Smallest value [1-3] Population variance 2  ( X   ) 2 N [1-4] Population standard deviation  ( X   ) 2 N [1-5] Sample variance s2  ( X  X ) 2 n 1 [1-6] Sample standard deviation s ( X  X ) 2 n 1 [1-7] Standard normal value z Standard error of mean x  z-value,  and  known z X  [2-1]   [3-1] n X  [3-2]  n Confidence interval for  , with  known X z  Confidence interval for  , with  unknown Testing a mean,  known X t z [4-1] n X   s n [4-2] [5-1] n BUS105e Copyright © 2015 SIM University Examination – July Semester 2015 Page 9 of 12 Testing a mean,  unknown t X  s [5-2] n Sample proportion BUS105e Copyright © 2015 SIM University Examination – July Semester 2015 p X n [6-1] Page 10 of 12 Appendix B: Statistical Tables AREA UNDER THE NORMAL CURVE Area of 0.4750 0 Z = 1.96 Example: To find the area under the curve between the mean and a point 1.96 standard deviations to the right of the mean, look up the value opposite 1.9 and under 0.06 in the table; 0.4750 of the area under the curve lies between the mean and a z value of 1.96. z 0.00 0.01 0.02 0.03 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 0.0000 0.0398 0.0793 0.1179 0.1554 0.1915 0.2257 0.2580 0.2881 0.3159 0.3413 0.3643 0.3849 0.4032 0.4192 0.4332 0.4452 0.4554 0.4641 0.4713 0.4772 0.4821 0.4861 0.4893 0.4918 0.4938 0.4953 0.4965 0.4974 0.4981 0.4987 0.0040 0.0438 0.0832 0.1217 0.1591 0.1950 0.2291 0.2611 0.2910 0.3186 0.3438 0.3665 0.3869 0.4049 0.4207 0.4345 0.4463 0.4564 0.4649 0.4719 0.4778 0.4826 0.4864 0.4896 0.4920 0.4940 0.4955 0.4966 0.4975 0.4982 0.4987 0.0080 0.0478 0.0871 0.1255 0.1628 0.1985 0.2324 0.2642 0.2939 0.3212 0.3461 0.3686 0.3888 0.4066 0.4222 0.4357 0.4474 0.4573 0.4656 0.4726 0.4783 0.4830 0.4868 0.4898 0.4922 0.4941 0.4956 0.4967 0.4976 0.4982 0.4987 0.0120 0.0517 0.0910 0.1293 0.1664 0.2019 0.2357 0.2673 0.2967 0.3238 0.3485 0.3708 0.3907 0.4082 0.4236 0.4370 0.4484 0.4582 0.4664 0.4732 0.4788 0.4834 0.4871 0.4901 0.4925 0.4943 0.4957 0.4968 0.4977 0.4983 0.4988 0.04 0.05 0.0160 0.0199 0.0557 0.0596 0.0948 0.0987 0.1331 0.1368 0.1700 0.1736 0.2054 0.2088 0.2389 0.2422 0.2704 0.2734 0.2995 0.3023 0.3264 0.3289 0.3508 0.3531 0.3729 0.3749 0.3925 0.3944 0.4099 0.4115 0.4251 0.4265 0.4382 0.4394 0.4495 * 0.4505 0.4591 0.4599 0.4671 0.4678 0.4738 0.4744 0.4793 0.4798 0.4838 0.4842 0.4875 0.4878 0.4904 0.4906 0.4927 0.4929 0.4945 0.4946 0.4959 0.4960 0.4969 0.4970 0.4977 0.4978 0.4984 0.4984 0.4988 0.4989 0.06 0.0239 0.0636 0.1026 0.1406 0.1772 0.2123 0.2454 0.2764 0.3051 0.3315 0.3554 0.3770 0.3962 0.4131 0.4279 0.4406 0.4515 0.4608 0.4686 0.4750 0.4803 0.4846 0.4881 0.4909 0.4931 0.4948 0.4961 0.4971 0.4979 0.4985 0.4989 0.07 0.08 0.0279 0.0319 0.0675 0.0714 0.1064 0.1103 0.1443 0.1480 0.1808 0.1844 0.2157 0.2190 0.2486 0.2517 0.2794 0.2823 0.3078 0.3106 0.3340 0.3365 0.3577 0.3599 0.3790 0.3810 0.3980 0.3997 0.4147 0.4162 0.4292 0.4306 0.4418 0.4429 0.4525 0.4535 0.4616 0.4625 0.4693 0.4699 0.4756 0.4761 0.4808 0.4812 0.4850 0.4854 0.4884 0.4887 0.4911 0.4913 0.4932 0.4934 0.4949 * 0.4951 0.4962 0.4963 0.4972 0.4973 0.4979 0.4980 0.4985 0.4986 0.4989 0.4990 0.09 0.0359 0.0753 0.1141 0.1517 0.1879 0.2224 0.2549 0.2852 0.3133 0.3389 0.3621 0.3830 0.4015 0.4177 0.4319 0.4441 0.4545 0.4633 0.4706 0.4767 0.4817 0.4857 0.4890 0.4916 0.4936 0.4952 0.4964 0.4974 0.4981 0.4986 0.4990 NOTE: For values of z above 3.09, use 0.4999 for the area. BUS105e Copyright © 2015 SIM University Examination – July Semester 2015 Page 11 of 12 T-DISTRIBUTION Left tail Critical t-score (negative) Degrees of freedom 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 Large (Z) Two Right Critical t-score (positive) .005 (one tail) .01 (two tails) 63.657 9.925 5.841 4.604 4.032 3.707 3.500 3.355 3.250 3.169 3.106 3.054 3.012 2.977 2.947 2.921 2.898 2.878 2.861 2.845 2.831 2.819 2.807 2.797 2.787 2.779 2.771 2.763 2.756 2.575 BUS105e Copyright © 2015 SIM University Examination – July Semester 2015 .01 (one tail) .02 (two tails) 31.821 6.965 4.541 3.747 3.365 3.143 2.998 2.896 2.821 2.764 2.718 2.681 2.650 2.625 2.602 2.584 2.567 2.552 2.540 2.528 2.518 2.508 2.500 2.492 2.485 2.479 2.473 2.467 2.462 2.327 .025 (one tail) .05 (two tails) 12.706 4.303 3.182 2.776 2.571 2.447 2.365 2.306 2.262 2.228 2.201 2.179 2.160 2.145 2.132 2.120 2.110 2.101 2.093 2.086 2.080 2.074 2.069 2.064 2.060 2.056 2.052 2.048 2.045 1.960 Critical t-score (negative) Critical t-score (positive) .05 (one tail) .10 (two tails) 6.314 2.920 2.353 2.132 2.015 1.943 1.895 1.860 1.833 1.812 1.796 1.782 1.771 1.761 1.753 1.746 1.740 1.734 1.729 1.725 1.721 1.717 1.714 1.711 1.708 1.706 1.703 1.701 1.699 1.645 Page 12 of 12