应用统计案例分析.doc

Case Problem FORECASTING LOST SALES The Carlson Department Store suffered heavy damage when a hurricane struck on August 31,1996.The store was closed for 4 months (September 1996 through December 1996), and Carlson is now involved in a dispute with its insurance company concerning the amount of lost sales during the time the store was closed .Two key issues must be resolved :(1)the amount of sales Carlson would have made if the hurricane had not struck; and(2)whether Carlson entitled to any compensation for excess sales from increased business activity after the storm. More than $8 billion in federal disaster relief and insurance money came into the county, resulting in increased sales at department stores and numerous other businesses.Table 1 shows the sales data for the 48 months preceding the storm. The U.S. Department of Commerce also has published total sales for the 48 months preceding the storm for all department stores in the county, as well as the total sales in the county for 4 months the Carlson Department Store was closed. Table 2 shows these data. Management has asked you to analyze these data and develop estimates of the lost sales at the Carlson Department Store for the months of September through December 1996. Management also has asked you to determine whether a case can be made for excess storm-related sales during the same period. If such a case can be made, Carlson is entitled to compensation for excess sales it would have earned in addition to ordinary sales.Table 1 SALES FOR CARLSON DEPARTMENT STORE, SEPTEMBER 1992 THROUGH AUGUST 1996Month1992 1993199419951996Jan.Feb.Mae.Apr.May.June.July.Aug.Sept.Oct.Nov.Des.1.711.902.744.201.451.802.031.992.322.202.132.431.902.132.564.162.311.982.022.232.392.142.272.212.892.292.834.042.311.992.422.452.572.422.402.502.092.542.974.352.562.282.692.482.732.372.312.23 Table 2 DEPARTMENT STORE SALES FOR THE COUNTY, SEPTEMBER 1992 THROUGH AUGUST 1996Month1992 1993199419951996Jan.Feb.Mae.Apr.May.June.July.Aug.Sept.Oct.Nov.Des.55.856.471.4117.646.848.060.057.661.858.256.463.057.653.471.4114.046.848.659.458.260.655.251.058.849.854.665.4102.043.845.657.653.456.452.854.060.647.454.667.8100.248.051.657.658.260.057.057.661.869.075.085.2121.8Managerial ReportPrepare a report for the management of Carlson Department Store that summarizes your findings, forecasts, and recommendations. Include the following.1. An estimate of sales had these been no hurricane.2. An estimate of countywide department store sales had these been no hurricane.3. Use the countywide department store sales for September through December 1996 and the estimate in part (2) to make a case for or against excess storm-related sales.4. An estimate of lost sales for the Carlson Department Store for September through December 1996通过运用时间序列分析方法，我们对1992年到1996年Carlson商店和全县商店的月销售额进行了分析，从而找出Carlson商店在这场飓风中所受到的实际损失。下面计算在没有飓风的影响时Carlson商店和全县商店在1996.91996.12的销售额的预测值。在Carlson的销售中：直线趋势公式为Y=a + b x ，将48个月的销售额取算术平均数得a=2.456458,将第24和25个月即1994年8月和9月的中间作为中间定点，运用公式代入得b=0.009169.将1996年912月代表的时间序列分别代入公式得销售额的趋势值。然后我们运用季节指数分析的方法计算每年九月到十二月四个月的月指数。将趋势值与对应的月指数值相乘，得相应的预测值。时间时间序列趋势值月指数预测值1996.0924.52.68109986.5402442.32022961996.1025.52.69026886.4165312.32483631996.1126.52.69943797.1309792.62198961996.1227.52.708606146.817333.9767029即在没有飓风影响时Carlson商店销售额的预测值分别为：1996.09： 2.32022961996.10： 2.32483631996.11： 2.62198961996.12: 3.9767029在全县的销售中，运用同样的分析方法：直线趋势公式为Y=a + b x ，其中1994年8月和9月的中间作为中间定点，得a=60.5375, b=-0.13023. 将1996年912月代表的时间序列分别代入公式得销售额的趋势值。运用季节指数分析的方法计算每年九月到十二月四个月的月指数。将趋势值与对应的月指数值相乘，得相应的预测值。时间时间序列趋势值月指数预测值1996.0924.557.3468190.09896351.6688811996.1025.557.2165887.41247550.0144281996.1126.557.0863497.86981655.8702961996.1227.556.95611158.8018990.447379即在没有飓风影响时全县商店销售额的预测值分别为：1996.09 ：51.6688811996.10 ：50.0144281996.11 ：55.8702961996.12 ：90.447379我们可以看出，在没有飓风影响的情况下，1996年全县的销售额预测值为699.10748；在有飓风的情况下，实际销售额为802.8。并且1996.9-12全县的销售额实际值均较无飓风时多一些。相较于原来一年不如一年的情况，将1996年的数据和1995年作比较，销售额反而有15.6%的涨幅。我们可以初步假定，飓风给销售带来了一定的积极影响。但是，我们的假定是否有根据，这个差别到底是由飓风带来的还是销售额年与年之间的波动呢？为了检验这个差别是否在统计学上是显著有效的，我们建立一个置信度为95%的置信区间。选取的样本大小为1993到1995的实际年销售额（假设销售额是相对稳定的）。我们推测，飓风对当地的商店销售额产生了影响。通过置信指标计算的置信区间为648.76到786.44，而实际的销售额比区间上限更大一些。所以1996年的飓风对县里的销售额有积极影响。 既然Carlson商店由于飓风的影响未能营业，并且飓风对销售是有积极影响的，那么我们可以断言，Carlson商店在飓风中所造成的实际损失比上面我们预测的销售额还要多。由于飓风的影响，使得全县的各个商店1996年9月12月不具备时间指标的规律性。1996年九月到十二月全县的实际销售额在全年的比例上升到了43.7%。而基于Carlson的数据，他们平均年销售的61%发生在本年的前八个月。假设这个