MGT 3: Quantitative Methods in Business Session 5 – Assignment (20 points total) Q1: Fill in the missing joint and marginal frequencies in the contingency table below. (1 point) X x1 Y x2 y1 94 y2 90 119 250 Q2. Fill in the missing joint and marginal probabilities in the contingency table below. (1 point) X x1 Y y1 0.11 y2 0.54 x2 0.59 Q3: Using the completed contingency table below, solve for the probabilities expressed in parts a–h. B Y N Y 0.34 0.17 0.51 N 0.39 0.10 0.49 0.73 0.27 1 A © Ryan Wagner, 2020. Do not copy or distribute without permission. (Q3 parts a–h: ½ point each) 𝑷(𝑩 = 𝑵) b. 𝑷(𝑨 = 𝒀 ∩ 𝑩 = 𝒀) c. 𝑷(𝑨 = 𝑵 ∩ 𝑩 = 𝒀) d. 𝑷(𝑨 = 𝑵) a. 𝑷(𝑨 = 𝒀 | 𝑩 = 𝒀) f. 𝑷(𝑨 = 𝒀 | 𝑩 = 𝑵) g. 𝑷(𝑨 = 𝑵 | 𝑩 = 𝑵) h. 𝑷(𝑩 = 𝑵 | 𝑨 = 𝒀) e. Q4: The Small Business Association (SBA) is a government organization that supports entrepreneurs by essentially co-signing commercial loans made by banks. Lending money to small businesses is inherently risky, given that approximately 50% of all small businesses fail within 5 years of their founding, often defaulting on their loans in the process. Currently, about 54% of all applications for SBA loans are approved. Of course, lenders do their best to only approve applications from those businesses that appear to be the most likely to succeed. Even so, the rate of failure is nontrivial: Among those businesses approved for an SBA loan, about 17% go on to fail within 5 years. Let r.v. A represent whether a small business’ loan application is approved. A = {Yes, No} Let r.v. F represent whether a small business fails within 5 years of its founding. F = {Yes, No} a. Use the information provided to complete the contingency table below: (2 points) F Y N Y A N b. What is the probability that a business will succeed if it is approved for a loan? (1 point) c. What is the probability that a business will succeed if it is not approved for a loan? (1 point) © Ryan Wagner, 2020. Do not copy or distribute without permission. Q5. The WestSide Group is a consulting firm that caters mainly to the pharmaceutical industry. The HR department at TWG has noticed a worrying trend: the company’s hourly employees have been quitting at an increasing rate. In order to minimize the exodus of its workers, analysts in the HR department are attempting to identify the root causes of the increase in turnover so that upper management can intervene as necessary. HR is aware of one main anecdotal complaint about working at TWG: the hours are too long. In order to explore the idea that long hours are driving employees to quit, HR decides to examine the past 2 years’ worth of employee data. The data includes records of 320 employees, 93 of whom have quit the company. The analysts decide to define ‘long hours’ as an average of 55+ hours per week, while employees averaging less than 55 hours are considered to work ‘normal hours’. They find that among the employees who have quit, 66.7% averaged long hours. One analyst remarks that even if long hours are driving some employees to leave, it shouldn’t be a top concern for the company, because the majority of employees (57.81%) don’t work long hours in the first place. Let r.v. H represent the average number of hours logged on an employee’s timecard. H = {Normal, Long} Let r.v. Q represent whether an employee has quit the company or not. C = {Yes, No} a. Use the information provided to complete the contingency table below: (2 points) Quit Yes No Long Hours Normal b. What is the probability that an employee who works long hours will quit the company? (1 point) c. Given your answer to part b, what might you conclude about the idea that long hours are driving employees to quit the company? (1 point) © Ryan Wagner, 2020. Do not copy or distribute without permission. Q6. A credit agency has created a classification model to predict whether customers will default on a loan. (“Yes” = customer defaulted on a loan). The model’s confusion matrix is displayed below: Actual No Yes No 3353 1413 Yes 3283 5223 Predicted a. Compute the accuracy statistic for the model, and provide a 1-2 sentence interpretation of your result. (1 point) b. Determine whether class imbalance appears to be an issue in this model, and use your answer to comment on whether the accuracy statistic computed in part a appears to reflect an actual improvement in predictive power. (1 point) c. Compute the sensitivity (recall) statistic for the model, and provide a 1-2 sentence interpretation of your result. (1 point) d. Compute the positive predictive value (precision) statistic for the model, and provide a 1-2 sentence interpretation of your result. (1 point) e. Consider that the model’s prediction will determine whether the bank offers a loan to the customer. In this context, consider what each error type (false positive vs. false negative) actually means. In a brief paragraph, answer the following: What are the financial implications of each error type? Which error type do you think is worse, and why? (2 points) © Ryan Wagner, 2020. Do not copy or distribute without permission.
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