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A wholesale furniture retailer stores in-stock items at a large warehouse located in Florida. Several years ago, a fire destroyed the warehouse and all the furniture in it. After determining the fire was an accident, the retailer sought to recover costs by submitting a claim to its insurance company.
As is typical in a fire insurance policy of this type, the furniture retailer must provide the insurance company with an estimate of “lost” profit for the destroyed items. Retailers calculate profit margin in percentage form using the Gross Profit Factor (GPF). By definition, the GPF for a single sold item is the ratio of the profit to the item’s selling price measured as a percentage. That is,
Item GPF = (Profit / Sales Price) * 100%.
Of interest to both the retailer and the insurance company is the average GPF for all of the items in the warehouse. Since these furniture pieces were all destroyed, their eventual selling prices and profit values are obviously unknown. Consequently, the exact value of the average GPF for all of the warehouse items lost in the fire is unknown. The value can, however, be estimated.
One way to estimate the mean GPF of the destroyed items is to use the mean GPF of similar, recently sold items. The retailer had sold 3,005 furniture items in the year prior to the fire and kept paper invoices on these sales. Rather than calculate the mean GPF for all 3,005 items (the data were not computerized), the retailer sampled a total of 253 of the invoices and computed the mean GPF for these items as 50.8%. The retailer applied this average GPF to the costs of the furniture items destroyed in the fire to obtain an estimate of the “lost” profit.
According to experienced claims adjusters at the insurance company, the GPF for sale items of the type destroyed in the fire rarely exceeds 48%. Consequently, the estimate of 50.8% appeared to be unusually high. (A 1% increase in GPF for items of this type equates to, approximately, an additional $16,000 in profit.) Consequently, a dispute arose between the furniture retailer and the insurance company, and a lawsuit was filed. In one portion of the suit, the insurance company accused the retailer of fraudulently representing its sampling methodology. Rather than selecting a sample randomly, the retailer was accused of selecting an unusual number of “high profit” items from the population in order to increase the average GPF of the sample.
To support its claim of fraud, the insurance company hired a CPA firm to independently assess the retailer’s true GPF. Through the discovery process, the CPA firm legally obtained the paper invoiced for the entire population of 3,005 items sold the year before the fire and input the information in to a computer. The selling price, the profit, profit margin, and month sold for these 3,005 furniture items are available in the file Firefraud.
QUESTION 1: Find the probability that a single randomly selected item will have a GPF that exceeds 50.8%.
QUESTION 2: Now find the probability that a sample of size 253 would have a mean GPF that exceeds 50.8%
YOUR ASSIGNMENT: You are the statisticians retained by the CPA firm that the insurance company hired. Write a report in which you explain the problem under consideration, any important background, and all relevant data. Include answers to questions 1 and 2 in your report, being sure to completely explain what you found and how you found the values (your methodology will be important in the court’s decision about the accuracy of your findings). What would you recommend that the insurance company do next?
As is typical in a fire insurance policy of this type, the furniture retailer must provide the insurance company with an estimate of “lost” profit for the destroyed items. Retailers calculate profit margin in percentage form using the Gross Profit Factor (GPF). By definition, the GPF for a single sold item is the ratio of the profit to the item’s selling price measured as a percentage. That is,
Item GPF = (Profit / Sales Price) * 100%.
Of interest to both the retailer and the insurance company is the average GPF for all of the items in the warehouse. Since these furniture pieces were all destroyed, their eventual selling prices and profit values are obviously unknown. Consequently, the exact value of the average GPF for all of the warehouse items lost in the fire is unknown. The value can, however, be estimated.
One way to estimate the mean GPF of the destroyed items is to use the mean GPF of similar, recently sold items. The retailer had sold 3,005 furniture items in the year prior to the fire and kept paper invoices on these sales. Rather than calculate the mean GPF for all 3,005 items (the data were not computerized), the retailer sampled a total of 253 of the invoices and computed the mean GPF for these items as 50.8%. The retailer applied this average GPF to the costs of the furniture items destroyed in the fire to obtain an estimate of the “lost” profit.
According to experienced claims adjusters at the insurance company, the GPF for sale items of the type destroyed in the fire rarely exceeds 48%. Consequently, the estimate of 50.8% appeared to be unusually high. (A 1% increase in GPF for items of this type equates to, approximately, an additional $16,000 in profit.) Consequently, a dispute arose between the furniture retailer and the insurance company, and a lawsuit was filed. In one portion of the suit, the insurance company accused the retailer of fraudulently representing its sampling methodology. Rather than selecting a sample randomly, the retailer was accused of selecting an unusual number of “high profit” items from the population in order to increase the average GPF of the sample.
To support its claim of fraud, the insurance company hired a CPA firm to independently assess the retailer’s true GPF. Through the discovery process, the CPA firm legally obtained the paper invoiced for the entire population of 3,005 items sold the year before the fire and input the information in to a computer. The selling price, the profit, profit margin, and month sold for these 3,005 furniture items are available in the file Firefraud.
QUESTION 1: Find the probability that a single randomly selected item will have a GPF that exceeds 50.8%.
QUESTION 2: Now find the probability that a sample of size 253 would have a mean GPF that exceeds 50.8%
YOUR ASSIGNMENT: You are the statisticians retained by the CPA firm that the insurance company hired. Write a report in which you explain the problem under consideration, any important background, and all relevant data. Include answers to questions 1 and 2 in your report, being sure to completely explain what you found and how you found the values (your methodology will be important in the court’s decision about the accuracy of your findings). What would you recommend that the insurance company do next?