BEA140 Quantitative Methods Assignment 3 Semester 2, 2017 (Marks in total - 45) Due Date: 4:00pm Friday 6 October 2017 Contribution to Unit Assessment: 10% Students should ensure that they have read and understood the guidelines for assignment (attached) Question 1 (12 marks) Tasmania imports grain from other regions of Australia. There is a danger that seeds of species which are damaging environmentally and economically will be imported along with the grain shipments. Measures must be taken to manage the risks in this situation. The grain lots arrive in shipping containers. To determine whether the seeds are fit for their intended use, the container lots are sampled. The sampling process involves taking out a fixed amount (1 kg) of grain from each container, and inspecting for contaminant species and identifying them. Then seeds of species from a declared list are counted. If the number of weed seeds included is judged to form an unacceptable risk, the container lot will be re-directed to safer uses such as milling or heat processing (this will be less profitable for the shipper). If there are weed seeds present in the container, the probability of observing them depends on the true rate of inclusion. Assume for a particular shipment that the weed seeds are included 'randomly' at an average rate of 1 per kilogram throughout the container. a) What distribution best describes the number of weed seeds found in a kilogram sample? (2 marks) b) Let the random variable X be the number of seeds found in the 1kg sample. Calculate the probabilities of there being 0, 1, 2 or 3 weed seeds observed in the (1 kg) sample, i.e. P(X=0), P(X=1), P(X=2) and P(X=3). (2 marks) c) Given your calculations in part (b) what is the probability of observing 4 or more seeds in a shipment, i.e. P(X 4)? (1 mark) d) Go on to complete the table below for different rates of inclusion of 1, 2 or 4 seeds per kg. (You have done the first column in parts (b) and (c). Two values are included to help you.) (3 marks) Average rate of inclusion in container (seeds per kg) No of seeds detected in sample 0 1 2 3 0.3679 1 2 3 4 or more 0.3528 4 The instruction manual for inspection officers says that if 4 or more seeds are detected in the 1kg sample then the container will be redirected to safer uses. e) What is the probability of accepting a container if the inclusion rate is 4 seeds per kg in the container lot? (1 mark) f) If a rate of 2 weed seeds per kg in the container lot is regarded as an acceptable risk, what is the probability of redirecting a container with an inclusion rate of 2 seeds per kg, according to the rule in the manual? (1 mark) g) Suggest how the testing procedure could be changed to minimise the risk of false positives, i.e. rejecting a shipment that is acceptable, while not increasing the risk of false negatives. (2 marks) Question 2 (18 marks) The growth rate of abalone can be monitored using a statistic known as the specific growth rate (SGR). If the growth rate for abalone in a tank is too low, it is not economic to spend money and use resources to continue to grow the abalone. In the early stages, a sample of abalone is monitored to provide the information on which a decision will be based. There are two options: Option 1. discard this batch and start the process again with a new batch of abalone because the growth rate is too low; Option 2. allow the present stock to continue growing. Decision making rule. To make a decision, a sample of ten abalone is tagged, the mean SGR is determined for the tagged shellfish, and a decision is made based on the following rule: Choose Option 1 if the mean SGR is less than 0.54. Choose Option 2 if the mean SGR is over 0.54. Probabilities of choosing different options. From past observation, it is known that the pattern of SGR among individual abalone is well approximated by a Normal distribution with standard deviation 0.2 a) For the indicated mean values for SGR in the table below, complete the table to establish the probabilities that the decision rule stated above will lead to the selection of Option 1 or Option 2. (4 marks) Probabilities of selecting different options based on mean SGR for ten abalone for different mean levels of SGR for the tank Mean SGR for tank Probability Option 1 Option 2 0.60 0.64 0.68 0.72 Proportion and number above the minimum acceptable SGR level. The minimum acceptable level of SGR for an abalone to reach a commercially acceptable size at the end of the growing period is 0.55. b) Let the random variable Y be the SGR of an individual abalone. For each value of the 'mean SGR for the tank', compute the proportion of abalone in the tank which have an SGR level above the minimum acceptable level 0f 0.55, i.e. P(Y>0.55). (4 marks) Mean SGR for tank 0.60 0.64 0.68 0.72 P(Y>0.55) c) If the mean SGR in the tank is 0.64, what distribution best describes the number of abalone in a batch of 100 that achieve the minimal acceptable growth rate? (3 marks) d) Is it valid to approximate the distribution above with a normal distribution? If so what normal distribution would you use? (3 marks) Chances of making a profit. A profit cannot be made under Option 1 since the process is stopped before the abalone mature. Under Option 2, for a batch of 100 abalone to return a profit, at least seventy percent of the abalone must have SGR levels above the minimum acceptable value of 0.55. Probabilities of making a profit under option 2 based on mean SGR for ten abalone for different mean levels of SGR for the tank Mean SGR for tank 0.60 0.64 0.68 0.72 70 or more acceptable e) For batches of 100 abalone, compute the probabilities that the number of abalone is at least equal to 70 and use these values to complete the table below. (2 marks) f) Assuming that the mean SGR is 0.64 what is the overall probability of making a profit? g) Do you think that the decision rule used by the company is reasonable? (1 mark) (1 mark) Question 3 (15 marks) A factory has been using three different machines to produce ball bearings. The ball bearings can be used if they have a diameter between 9-11mm. The specs of the 3 machines are: New machine: = 10mm, = 0.5mm Old machine: = 10mm, = 1mm Borrowed machine: =9, = 0.3mm a) For each machine compute the probability of producing a useful ball bearing (6 marks) b) Overall the factory produces 70% of ball bearings on the new machine, 20% on the old machine and 10% on the borrowed machine. Fill in contingency table below to show joint and marginal probabilities for the events defined. (6 marks) New machine (N) Old machine (O) Borrowed machine (B) Useable (U) Non Useable c) Given the fact that a randomly selected ball bearing turns out to be usable, what is the probability that it was made on the new machine? (3 marks) Assessment Guidelines The rubric below provides a guideline for students of the expectations with respect to key assessment criteria. Criteria Applies To Identify and state any assumptions required to allow the use of a particular methodology. No Penalty Assumptions 1 Methodology Questions or question parts Evaluation Presentation Whole Assignment 1 2 Student shows understanding of the problem. Appropriate methodology chosen for the problem. Enough detail in the answer for the marker to follow the logic of the student's approach. (As an example, the level of detail provided in examples in lecture notes.) No Penalty Answers are correct, given the values that the student started the problem (or problem part) with. 2 No penalty. Interpretations, when asked for are correct and complete. Answers are feasible. No penalty Interpretation and reflection Timeliness Excellent Intermediate Poor Assumptions identified but misstated or misrepresented. Penalty 0-1 marks Failure to identify and state an assumption required to allow the use of a particular methodology, or use of a wrong assumption. Penalty 1-2 marks Student shows some understanding of the problem. General approach basically correct, but some steps wrong or inappropriate. Marker has some difficulty following student's approach and/or there are significant jumps of logic or missing details Penalty <= 50% of marks available. Minor calculation and/or transcription errors leading to incorrect answers. Penalty 1 mark. Major and/or repeated errors. Penalty 1-2 marks. Interpretations, when asked for, are partially correct or partially complete. Answer is NOT feasible but the student identifies and explains why the answer can't be right. Penalty 1 mark. Interpretations, when asked for are either well off the mark or are missing. Answer is NOT feasible but the student does not identify or explain this. Penalty 2 marks. Submitted as a \".docx\