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Quantitative Methods Examination L1AF105 202001 • The maximum mark for this paper is 100. Use black ink or ballpoint pen. • You should complete ten of the twelve questions only. • All necessary working should be shown, otherwise marks for method may be lost. • The duration of this exam is two hours. • Useful formulae and tables are given at the end of this paper ILSC Study Skills Incorporated and Tested Skills Yes/No Comments Presentation Skills yes Students should present their written work neatly and in an organized manner. Self - Directed Study yes Students will have needed to study independently to be prepared for the examination. Writing Skills yes (Accuracy, Coherence) Analysis and Problem and accurately. yes Solving Planning Aspects (Structure, Content Development) Students are tutored to present solutions laid out clearly Students are required to determine the correct statistical methods to solve the given problems. yes Students need to present solutions in a structured, logical manner. 1). a) Solve the following pairs of simultaneous equations: i) 3x + 5 y = 6 5 y − x = 12 ii) 3y − 9x − 5 = 0 3x + 9 y = 10 b) Simplify the following: 3 i) 𝑥(𝑥 2 ) × (√𝑥 4) 3 ii) ( −2 ) 𝑥 2 𝑥 c) Simplify the following: i) 3𝐿𝑜𝑔5 − 4𝐿𝑜𝑔3 + 2𝐿𝑜𝑔6 4 2 ii) 12 𝐿𝑜𝑔216 + 4𝐿𝑜𝑔3 − 4 𝐿𝑜𝑔36 [10] 2). a) The probability that Portsmouth will have a good weather is 0.1. A sample of 40 days within a year is analysed. Find the following, giving your answers for (ii) and (iii) correct to 3 significant figures: i) The expected number of days with good weather from the sample. ii) The probability that exactly 10 days will be days with good weather. iii) The probability that more than 4 days will have good weather. b) 𝑥 𝑃(𝑋 = 𝑥) i) ii) 2 4 6 8 10 0.2 0.2 0.4 a 0.05 Find 𝜇 and 𝜎 correct to 1 decimal place. ii) Find 𝑃(𝑋 ≥ 𝜇) [10] 3). In a call centre, the duration of a sample of phone calls is recorded. The results are shown below: Length of call,𝒍, (mins) Frequency 0.5 < 𝑙 ≤ 6.5 5 6.5 < 𝑙 ≤ 10.5 31 10.5 < 𝑙 ≤ 14.5 30 14.5 < 𝑙 ≤ 18.5 37 18.5 < 𝑙 ≤ 20.5 19 a) Calculate estimates, to two decimal places, for: i) The mean length of call times. ii) The standard deviation in the length of call times. iii) The median length of call times. b) Using your answers to a) calculate Pearson’s Coefficient of Skewness. Given your answer, would you say that the mean or the median would be the best measure to represent the length of call times? [10] 4). A small café is self-service and there is only one till to serve customers. Customers select the items they want, then join a queue at the till and wait to receive service from the till operator. As there are sometimes customers waiting during busy periods, an investigation is conducted to see if it is worth installing a second till. A survey is carried out to find the time between customer arrivals (inter-arrival time) and how long it takes for each customer to be served at the till (service time), to the nearest minute. The results are below: Inter-arrival time (nearest minute) 1 2 3 4 5 Probability (%) 15 27 31 21 6 Service time (nearest minute) 2 3 4 5 6 Probability (%) 12 39 26 15 8 a) Perform a simulation for 10 customers, with the assumption that the first customer arrives at time zero. Copy and complete the 3 tables below to show the simulation: Inter-arrival time (nearest minute) 1 2 3 4 5 Probability (%) 15 27 31 21 6 Service time (nearest minute) 2 3 4 5 6 Probability (%) 12 39 26 15 8 Random numbers Random numbers Cust RN Interarrival time 1 0 Customer Arrives Start service RN Service time Service end Wait in queue Queue length (max) 35 2 42 92 3 22 78 4 03 98 5 80 16 6 19 06 7 37 31 8 62 34 9 93 82 10 55 17 b) In your simulation, what is the maximum number of customers waiting and the maximum time spent queuing? Do you think installing a second till to serve customers is justified? Why/why not? [10] 5). A study has been carried out to examine the hair length of women within the UK. A large sample of women was studied, and it was found that the sample was normally distributed with a mean of 30cm and a standard deviation of 8cm. Give your answers to the following questions correct to 3 decimal places. a) The participants are selected at random. Find the probability that the women’s hair is: i. Longer than 40cm ii. Shorter than 36 cm iii. Is between 24 and 32 cm b) Between what lengths would you expect 68%, 95% and 99.7% of women’s hair length to be? [10] 6). Laminate floor boards are to be fitted throughout an office. The precedence table below shows the tasks involved and their duration. Activity A: Remove carpets Duration (days) 0.75 Immediate Predecessors - B: C: D: E: Lay polythene Lay foam Lay boards Paint skirting 1 0.9 3 2 A B C A F: Stain beading G: Fit beading H: Fit doorplates 1 1.3 1 D, E, F D I: Clean 0.9 G, H a) Complete an activity network for the project. b) Find the earliest start time and latest finish time for each activity. c) Write down the critical activities. d) Due to workers being ill, Activity E is delayed by 2.5 days and Activity G is delayed by 1 day. What effect does this have on the completion of the project? Give your reasoning. [10] 7). GlobalPharma’s total estimated revenue from the sale of x units of a vaccine is given by 𝑅(𝑥) = 𝑥 2 − 5𝑥 + 10 Find a) the average revenue b) the marginal revenue c) the marginal revenue at x = 25 units d) the actual revenue from selling the 51st item. [10] 8). GlobalPharma sets up a factory to manufacture a vaccine with a fixed cost of £150,000. The variable cost of the vaccine is estimated at 20% of its selling price, when the product sells at the rate of £30 per unit. a. Find the revenue if 30,000 vaccines are sold. b. Find the total cost if 30,000 vaccines are sold. c. Find the break-even point. d. If GlobalPharma decreased the selling price to £25 per unit, what would the new break-even point be? [10] 9). A small company has two salesmen. The management wants to know if the mean number of sales per month for these salesmen are different. It is known that the monthly variance in sales for the first salesman is 1900 and for the second salesman is 12100. A sample of 12 months for the first salesman gives the following monthly sales numbers: 300 313 350 454 379 300 355 387 368 419 384 488 A sample of 15 months for the second salesman gives the following monthly sales numbers: 499 380 375 490 399 354 427 471 354 280 416 466 327 389 425 Test at the 5% significance level whether the mean daily sales for these two salesmen are different. You may assume that the sales are normally distributed. [10] 10). A factory makes two types of car parts, type X and type Y. Each type X part takes 10 hours to make and each type Y part takes 12 hours to make. In each week there are 200 hours available to make car parts. To satisfy customer demand, at least 5 of each type of car part must be made each week. When completed, the car parts are put into one container for shipping. The volume of the container is 7 m3. A type X car part occupies a volume of 0.5 m3 and a type Y car part occupies a volume of 0.3 m3. The four inequalities derived from the above information are shown on the graph below, and are labelled A, B, C and D. y A B C D x a) If 𝑥 represents the number of type X car parts produced, and 𝑦 represents the number of type Y car parts produced, write the inequalities: A B C D b) The profit on each type X car part is £100 and on each type Y car part is £70. The weekly profit is to be maximised. Write down the objective function and find the maximum profit. [10] 11). a) A random sample of 100 students are measured and found to have a mean height of 167.7cm, with a standard deviation of 7cm. The population can be assumed to be normally distributed. Find the 95% and 98% confidence intervals for the actual population mean. b) A sample of monthly rental prices for student accommodation in Portsmouth are shown below. Assuming that rental prices are normally distributed, find 95% and 99% confidence intervals for the mean monthly rental cost in the city. £750 £755 £695 £750 £770 £700 £770 £760 £725 £670 £780. [10] 12). a) GlobalPharma manufactures a vaccine with total cost function given by: 𝐶(𝑥) = 𝑥 3 − 200𝑥 2 + 10000𝑥 + 6000 where x is the number of units produced. Determine the number of units that must be produced to minimise the total cost. b) GlobalPharma charges £550 per unit for a different product, for an order of 40 units or less. The charge is reduced by £5 for each unit ordered in excess of 40 units. Find the size of the largest order GlobalPharma should allow, to receive a maximum revenue. [10] END OF QUESTIONS USEFUL FORMULAE AND TABLES 𝑥̅ = ∑𝑥 𝑥̅ = 𝑛 ∑ 𝑓𝑥 𝑥−𝜇 𝑡=𝑠 ∑𝑓 ⁄ 𝑛 √ 𝑚𝑒𝑑𝑖𝑎𝑛−𝐿𝐶𝐵 Grouped data: 𝑈𝐶𝐵−𝐿𝐶𝐵 = 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑖𝑙𝑒−𝐿𝐶𝐵 𝑈𝐶𝐵−𝐿𝐶𝐵 𝑚𝑒𝑑𝑖𝑎𝑛 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛−𝐿𝐶𝐵 𝐶𝑢𝑚 𝑓𝑟𝑒𝑞 𝑈𝐶𝐵 𝐶𝑢𝑚 𝑓𝑟𝑒𝑞 −𝐿𝐶𝐵 𝐶𝑢𝑚 𝑓𝑟𝑒𝑞 = 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑖𝑙𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛−𝐿𝐶𝐵 𝐶𝑢𝑚 𝑓𝑟𝑒𝑞 𝑈𝐶𝐵 𝐶𝑢𝑚 𝑓𝑟𝑒𝑞 −𝐿𝐶𝐵 𝐶𝑢𝑚 𝑓𝑟𝑒𝑞 𝑆𝑎𝑚𝑝𝑙𝑒 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒: 𝑠 2 = Ungrouped data: ∑(𝑥−𝑥̅ )2 𝑛−1 𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒: 𝜎 2 = 𝑆𝑎𝑚𝑝𝑙𝑒 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒: 𝑠 2 = Grouped data: 𝑄𝑢𝑎𝑟𝑡𝑖𝑙𝑒 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑠𝑘𝑒𝑤𝑛𝑒𝑠𝑠 = 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑣𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛 = Discrete random variables: ∑(𝑥−𝜇)2 𝑁 ∑ 𝑥2 𝑛−1 , OR 𝜎 2 = ∑ 𝑓(𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡−𝑥̅ )2 𝑛−1 𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒: 𝜎 2 = 𝑃𝑒𝑎𝑟𝑠𝑜𝑛′ 𝑠 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑠𝑘𝑒𝑤𝑛𝑒𝑠𝑠 = , OR 𝑠 2 = − (∑ 𝑥)2 𝑛(𝑛−1) ∑ 𝑥2 𝑁 , OR 𝑠 2 = ∑ 𝑓(𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡−𝜇)2 𝑁 − 𝜇2 ∑ 𝑓(𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡)2 , OR 𝜎 2 = 𝑛−1 ∑ 𝑓(𝑚𝑖𝑑𝑝𝑜𝑛𝑡)2 𝑁 (𝑄3 − 𝑄2 ) − (𝑄2 − 𝑄1 ) (𝑄3 − 𝑄1 ) 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 × 100 𝑀𝑒𝑎𝑛 𝑉𝐴𝑅(𝑋) = 𝐸(𝑋 2 ) − [𝐸(𝑋)]2 𝐸(𝑋) = ∑ 𝑥 𝑃(𝑋 = 𝑥) 𝑥−𝜇 𝜎 Confidence intervals: − 𝜇2 3(𝑚𝑒𝑎𝑛 − 𝑚𝑒𝑑𝑖𝑎𝑛) 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑋~𝐵𝑖𝑛(𝑛, 𝑝): 𝑃(𝑋 = 𝑟) = 𝑛𝐶𝑟 𝑝𝑟 𝑞 𝑛−𝑟 𝑋~𝑁(𝜇, 𝜎 2 ): 𝑧 = (∑ 𝑓𝑥)2 − 𝑛(𝑛−1) 𝜇 = 𝑥̅ ± 𝑧 𝜎 √𝑛 , 𝜇 = 𝑥̅ ± 𝑧 𝑠 √𝑛 , 𝜇 = 𝑥̅ ± 𝑡 𝑠 √𝑛 2 2 1 2 𝑠 𝑠 For 𝑋̅1 − 𝑋̅2 ~𝑁 (0, 𝑛1 + 𝑛2 ): 𝑧= (𝑋̅1 −𝑋̅2 )−0 2 2 𝑠 𝑠 √ 1+ 2 𝑛1 𝑛2 Product moment correlation coefficient: Least squares regression line: 𝑟= 𝑦 = 𝑎 + 𝑏𝑥 ∑(𝑥−𝑥̅ )(𝑦−𝑦̅) √[∑(𝑥−𝑥̅ )2 (𝑦−𝑦̅)2 ] where 𝑏 = OR ∑(𝑥−𝑥̅ )(𝑦−𝑦̅) ∑(𝑥−𝑥̅ )2 𝑟= (∑ 𝑥 ∑ 𝑦) 𝑛 2 (∑ (∑ 𝑦)2 𝑥) √(∑ 𝑥 2 − )(∑ 𝑦 2 − ) 𝑛 𝑛 ∑ 𝑥𝑦− and 𝑎 = 𝑦̅ − 𝑏𝑥̅ P a g e | 12 Quantitative Methods L1AF105 Final Examination 202001 ...
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