Question 2 (16 marks) a. A consultancy firm took a random sample of readers of a motoring website across Australia to investigate the preferred size of sporting vehicle by location. The table below provides counts from the data set for the variables Vehicle Size and Location. Use the table to answer the following questions. Location 4WD SUV Grand total City 62 88 150 Suburbs 73 92 165 Rural 63 87 150 Grand total 198 267 165 i. Determine the probability that a reader in the sample prefers an SUV; (1 mark) ii. Determine the probability that a reader prefers an SUV, given that the person is from a Rural location; and (1 mark) Hi. By considering the prior two probabilities, say if it appears that the variable Vehicle size is independent of the variable Location. Justify your reasoning. (2 marks) b. The annual yield (expressed as a percentage) for all investments in a particular asset class in the latest financial year was normally distributed with a mean of 3.5% and a standard deviation of 1.05%. i. Represent the probability that a randomly selected investment in this class will have an annual yield greater than 5.00% on a diagram and determine this probability. (4 marks) ii. Represent the value for annual yield that will determine the lowest 5% of investments in this class exceed on a diagram and determine this value. (3 marks) c. A steel fabricator fills 60% of its orders from its factory in Albury and the remainder from its factory in Wagga. The fabricator finds that 3% of Albury orders are completed late, while this increases to 7% completed late for the Wagga factory. In your analysis below use the notation A to represent the event that an order is filled from Albury and [ to represent the event that a job is late. i. Determine the probability that an order was filled by the Wagga factory, that is P(A) (1 mark) ii. Represent the situation described as a probability tree using the notation provided and with outcomes and probabilities labelled. (2 marks) iii. Identify branches on your probability tree where the order is late and then calculate the proportion of all orders that are late. (2 marks)