Pennsylvania Profit Centre Data Analysis - Essay Example

?Data Analysis Table of Contents Data Analysis Table of Contents 2 3 The present paper discusses the way that HH Industries had been running their business over a year. It attempts to assess their trends in customer sales and orders received in addition to the way that their machines are functioning. The industry consists of three profit centers based in Florida, Arizona and Pennsylvania whose operations are being examined separately. 3 Introduction 3 Methods 3 Results, Conclusions and Recommendations 4 Answer to Question 1 4 Answer to Question 2 8 Answer to Question 3 10 Answer to Question 4 13 Answer to Question 5 14 Answer to Question 6 15 Answer to Question 7 16 Answer to Question 8 17 General Discussion 18 References 19 Bibliography 19 Anderson, D. R., Sweeney, D. J. & Williams, T. A. (2008). Essentials of Modern Business Statistics (4th Edition). Ohio: South-Western Cengage. 19 Abstract The present paper discusses the way that HH Industries had been running their business over a year. It attempts to assess their trends in customer sales and orders received in addition to the way that their machines are functioning. The industry consists of three profit centers based in Florida, Arizona and Pennsylvania whose operations are being examined separately. Introduction The situation prevailing over HH Industries is examined in the present case. It tries to assess the summary statistics for variables such as customer purchases and sizes of order for all three component profit centers as well as the entire company. This is the way through which it attempts to examine the trends that the company is taking over time. Methods In order to attain the objectives of the paper, central tendency methods and probability distributions have been used with the help of MS-Excel. Results, Conclusions and Recommendations Answer to Question 1 a) The relative frequency distributions and histograms for the company’s daily average order size in the first and second quarters have been presented underneath – For Quarter 1 For Quarter 2 Histogram representing of the company’s daily average order size for Quarter 1 Histogram representing of the company’s daily average order size for Quarter 2 b) The following charts display the histograms representing daily total orders of the company, for each quarter. Frequency of the company’s daily orders around the 3rd quarter of 1990 Frequency of the company’s daily orders around the 4th quarter of 1990 Frequency of the company’s daily orders around the 1st quarter of 1991 Frequency of the company’s daily orders around the 2nd quarter of 1991 c) Each quarter is found to be displaying almost equivalent trends in terms of daily orders received by the company. During the 3rd quarter of 1990, the company’s daily orders had been quite high and uniformly distributed. This implies that the proximity in the location of mean, median and mode in this case. However, this uniformity is visible more for the lower amounts of orders in contrast to that of the 1st quarter of 1991 which shows uniformity for the higher orders. On the other hand, this consistency cannot be noted in the 4th quarter of 1990. The 2nd quarter of 1991 on the other hand displays a consistency in the number of orders received throughout the period. A possible conclusion which could be drawn from the above statements is that the people are instigated to purchase more frequently over the years gradually as the company attains more and more popularity with them. It might also be possible that the customers to the company have realized the higher expenditure involved in storing inventories which is why they have shifted towards making smaller purchases frequently. Answer to Question 2 a) The central tendency measures computed for the quarterly data on number of orders and average order size have been presented in the following table – Central Tendency Measures Quarter 3 of 1990 Quarter 4 of 1990 Quarter 1 of 1991 Quarter 2 of 1991 Daily orders Average order size Daily orders Average order size Daily orders Average order size Daily orders Average order size Mean 155.8254 149.5126 171.6719 120.6767 171.2581 126.0114 177.8125 119.9309 Median 156 132.9831 168.5 120.0882 171.5 125.7397 177 116.7149 Mode 150 #N/A 163 #N/A 177 #N/A 191 #N/A Laura had intuitively concluded that the daily number of orders received would be higher due to greater popularity among the people. However, the average order sizes grow smaller over time as people realize their losses in storing inventories of the products. The above table is in alignment to the conclusions drawn by Laura in this context as the mean number of daily orders measured between the 3rd quarter of 1990 and 2nd quarter of 1991 tend to get higher while the average sizes of orders become smaller simultaneously. A similar conclusion might be drawn from the median values as well. The modes of the variables however, are inconclusive for average order size though if they are considered for the daily order sizes, they arrive at a similar conclusion as well. Mean is the most appropriate measure of central tendency in this regard given that the data values do not exhibit extremely high values for either of the two variables and rather tend towards uniformity. Moreover, there are little chances of any customers buying any specific amount of the commodity in question so that there are high chances of repetition (Huat et. al., 2008, p. 113). The company’s total sales for all the four quarters have been presented in the table below – Quarter Total Sales ($) Quarter 3 of 1990 1482788 Quarter 4 of 1990 1327508 Quarter 1 of 1991 1346084 Quarter 2 of 1991 1368181 Total sales volume are actually found to be the highest for the 3rd quarter of 1990, but it started deteriorating after that though eventually the figures started improving with time. Hence, Stanley’s assumption that the total sales are doing well could actually be regarded as true. b) The mean number of daily orders and order size for Profit Centre 3 (Pennsylvania) over the last four quarters are 28.17 and 99.013 respectively. A comparison between the daily orders received by Pennsylvania Profit Centre and the entire company has been displayed in the following graph. It shows that the trends in the number of orders received by Pennsylvania Profit Centre matches with that of the entire company. However, during the initial period, i.e., a considerable part of the 3rd quarter of 1990, the daily order sizes had been zero for Profit Centre 3. The remaining period on the other hand, shows considerable amount degree of alliance in trends for the two entities. However, daily order sizes of Pennsylvania Profit Centre and the entire company are not found to match in their trends. For instance, in the initial period, the daily order sizes of the former had been nil, in contrast to those of the whole group which exhibited the peak order sizer during that phase. On the other hand, when Profit Centre 3 exhibited the highest order size, that of the entire company had been experiencing one of its lowest order sizes as the bar diagram underneath shows. Comparing the trends of number of orders and average order sizes received by Pennsylvania Profit Centre and the entire company indicates that it had been a rather wise decision to investigate the performance of each profit centre separately, by Laurel. Answer to Question 3 a) The following table helps to draw a comparison between the inter quartile range and the actual ranges in average order sizes for each of the four quarters, have been presented in the underlying table. Quarters 3rd Quartile 1st Quartile Inter quartile Range Maximum Minimum Range 1990:Q3 155.882 116.9412 38.9407221 1113.611 66.17576 1047.436 1990:Q4 133.1664 106.3325 26.83394269 190.8071 69.96154 120.8456 1991:Q1 142.0064 108.9314 33.0750239 202.065 76.41875 125.6463 1991:Q2 127.2414 107.0388 20.20251257 197.0058 76.77439 120.2314 The above table shows huge differences in inter quartile range and the total range of the variables for each one of the four quarters. The difference is especially stark for the 3rd quarter of 1990. In the remaining cases the difference is found to be much lower. b) The quarterly sample variance, standard deviation values and coefficient of variation for number of orders has been presented in the following table – Number of Orders Quarters Sample variance Standard deviation Mean Coefficient of variation 1990:Q3 281.3399898 16.77319259 155.8253968 0.107641 1990:Q4 467.7160218 21.62674321 171.671875 0.125977 1991:Q1 548.3913273 23.41775667 171.2580645 0.13674 1991:Q2 336.9484127 18.35615463 177.8125 0.103233 Similarly, the quarterly sample variance, standard deviation values and coefficient of variation for the average order size has been presented in the following table – Average size of order Quarters Sample variance Standard deviation Mean Coefficient of variation 1990:Q3 16007.47504 126.5206506 149.5126249 0.846221 1990:Q4 582.4084767 24.13314063 120.6766512 0.199982 1991:Q1 702.6808024 26.5081271 126.0114103 0.210363 1991:Q2 597.0812627 24.43524632 119.9308558 0.203744 The coefficients of variation show the degree of variations in number of orders and average order size for each one of the four quarters. They depict the number of orders to vary the highest during the first quarter of 1991 while the lowest deviation is found for the consecutive quarter, i.e., second quarter of 1991. On the other hand, the degree of variation is found to be the highest for average order size during the 3rd quarter of 1990, owing to the peak attained during the initial part of 3rd quarter. The lowest variation is reported for the first quarter of 1991. c) The coefficient of variation for both the number of orders and average order size for the entire 12-month period, separately for each of the three profit centers has been provided underneath as follows – ORDERS 1 SALES 1 ORDERS 2 SALES 2 ORDERS 3 SALES 3 Standard Deviation 15.83767 12079.19 9.967258 1530.714 11.88932 1494.553 Mean 95.41897 15085.96 45.58893 3970.028 28.16996 2780.221 Coefficient of variation 0.16598 0.800691 0.218633 0.385568 0.422057 0.537566 The three profit centers display wide difference with each other. The number of orders received is found to be varying the highest for profit centre 1 while it is the lowest for that of profit centre 2. However, the degree of variation is not found to be too high between the numbers of orders received by each of the three profit centers. On the other hand, the average sales of these three profit centers are found to vary highly amongst themselves. It is the highest for profit centre 1 and the lowest for profit centre 3. However, the degree of deviation of sales volume for profit centre 2 is found to be closer to that of profit centre 3. Thus it might also be said that the high sales volume attained by the company during the initial phases had possibly been the result of the sales made by profit centre 1. d) The above findings could be followed by the following recommendations to the staff – Profit centre 1 could be treated as an ideal by other profit centers given that it experiences a higher mean order but lower degree of variation in the same. On the other hand, it shows the highest degree of inconsistency in terms of total sales. Hence, it might also be advised that while collecting data, the outlier points with extreme values must be omitted. Answer to Question 4 a) It had been found that the number of times that Machine 1 went down in 250 trials had been 27 while the number of times that Machine 2 had been down in 250 trials had been 27 as well. Moreover, the events that either machine will be down on a particular day are independent of each other. Thus, P (Machine 1 had been down) = 27/250 and P (Machine 2 had been down) = 27/250 So, the probability that a machine will be down on any particular day will be, P (Machine 1 had been down U Machine 2 had been down) = 27/250 + 27/250 – (27/250 x 27/250) = 0.204336 b) With 250 working days per year i. The probability that any one machine out of two will be down through the year will be, (250 x 0.204336) = 51.084 ? 51 ii. The probability that both machines will be down on a particular day will be, P (Machine 1 had been down П Machine 2 had been down) = 27/250 x 27/250 = 0.011664 Hence, out of 250 days, number of days when both machines could be expected to stay down will be, (250 x 0.011664) = 2.916 ? 3 days. Answer to Question 5 The following data is necessary to calculate the expected yearly cost in the current situation – Proposal 1 Lease price of two copiers per month = $ 350. Thus, the price of two leasing two copiers per year = $ (350 x 12) = $ 4200 Probability of a leased machine being down on any day = 0.05, Service call for two machines = $ 100 Proposal 2 Purchasing a new state-of-the-art machine to replace both old copiers = $ 8,750 During a period of one year, the service call will be free of cost for the new machine. However, after a year, the service call charge will be $ 175 per call. Moreover, probability of the machine being down = 0.017 Hence, the expected yearly costs are bound to be different for the period after one year due to the additional service call charges being incorporated. Moreover, the probability that any one of the two proposals would be granted might be regarded as ?, i.e., 0.5. Hence, the excepted yearly cost could be calculated as follows – Expected yearly cost for the 1st year = 0.5 [{350 + (0.05 x 100 x 30)} x 12] + 0.5 [8750] = $ 7,375 On the other hand, expected yearly cost for the periods past the first year will be, 0.5 [{350 + (0.05 x 100 x 30)} x 12] + 0.5 [0.017 x 175] = $ 3001.49 Answer to Question 6 In a period of 3 years, accepting either one of the two proposals could reap the following results that could be presented in the following tabular forms – Proposal 2 Cost of purchase Service Cost to be incurred per month Yearly Expenses Year 1 8750 0 8750 Year 2 0 89.25 89.25 Year 3 0 89.25 89.25 Total Cost 8928.5 It had been assumed here that the number of days in a month is 30 so that the service cost likely to be incurred each month has been calculated in compliance to the fact in cases of either ones of the proposals. A three year analysis is necessary in this case as assessing on the basis of the first year might be misleading. During the first year Proposal 1 is found to be much less expensive than Proposal 2. However, the consecutive years find Proposal 2 to be costing much lower than Proposal 1. In fact, at the end of 3 years, if time-value of money is ignored, Proposal 2 is found to be much more profitable than Proposal 1 and thus, is the better alternative. Answer to Question 7 a) The average number of calls received per hour has been presented in the table underneath – b) The present case could be solved with the help of Poisson distribution. However, the problem is not to determine the probability for the same but to figure out the number of cases, i.e., the number of representatives to be recommended. Here, P (x, µ) = 0.98, µ = Mean = 8 and x = to be calculated Hence, according to the Poisson formula, 0.98 = (e-8. 8x)/ x! = (0.000335 x 8x)/ x! => 8x/ x! = 2921.323 Using Solver for the purpose, the following solution have been reached at, x = 9.99995 ? 10 Hence, the number of sales representatives that Laurel and Stanley should recommend is 10. c) It had been recommended that for attending 8 calls per hour, 10 sales representatives are needed. However, Stanley handles 2 calls an hour which implies that the number of sales representatives must necessarily be high so as to match the criteria mentioned above. Answer to Question 8 a) To figure out the distribution of customers’ purchases, it is necessary to derive the histogram of the same, which has been represented as follows – The distribution could be considered as following a normal distribution owing to the fact that the mean and median are nearer to each other. b) The summary statistics for the number of customer purchases has been tabulated as follows – Summary Statistics Mean 13705.8875 Median 13291.5 Standard Deviation 4396.79134 c) Assuming that the active customer accounts are normally distributed with mean = 13705.8875 and standard deviation = 4396.79134, the following statistics could be calculated as under. i. Here, X = 20,000 Thus, the corresponding standard normal value will be, Z = (20000 - 13705.8875)/ 4396.79134 = 1.43 Hence, P (X > 20000) = P (Z > 1.43) = 1 – P (Z < 1.43) = 1 - 0.92386 = 0.076 ii. Here, X = 10,000 Thus, the corresponding standard normal value will be, Z = (10000 - 13705.8875)/ 4396.79134 = -0.84286 Hence, P (X < 10000) = P (Z < -0.84286) = 0.19965 iii. According to the data being provided in Table 4.0, 93.75% of the customers make purchases less than $ 20,000, implying that 6.25% of the people actually make purchases worth more than $ 20,000. On the other 18.75% of the customers are actually found to be making purchases less than $ 10,000 in contrast to the calculated 19.96%. General Discussion The above section suggests that the company had been faring well through the four quarters with eventual improvements. However, it might not be said that all three profit centers had been faring equally well over the years. For instance, Pennsylvania Profit Centre had not been performing up to the mark. On the other hand, Arizona could rather be treated as an ideal in this respect. Thus, it would be vital to assess each of them separately. References Huat, O. S. et al. (2008). Additional Mathematics. Malaysia: Pelangi Publishing Group Bhd. Bibliography Anderson, D. R., Sweeney, D. J. & Williams, T. A. (2008). Essentials of Modern Business Statistics (4th Edition). Ohio: South-Western Cengage. Black, K. (2009). Business Statistics. London, UK: Wiley. Bulmer, M. G. (1979). Principles of statistics. USA: Dover. Chiang, C. L. (2003). Statistical methods of analysis. USA: World Scientific. Cohen, J. Y. (2008). Statistics and data with R: an applied approach through examples. London, UK: Wiley. Downing, D. & Clark, J. (2010). Business Statistics. London, UK: Barron’s. Gordon, D. (2005) “Metrics Matter”, The Thomson Corporation and National Mortgage News. Available at http://www.freddiemac.com/hve/pdf/dougwhitepaper_metricsmatter.pdf. Gravetter, F. J. & Wallnau, L. B. (2008). Statistics for the Behavioral Sciences. USA: Cengage Learning. Ott, L. & Longnecker, M. (2008). An introduction to statistical methods and data analysis. USA: Cengage Learning. 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