1. Can you identify holiday periods or special events that cause the spikes in the
data?
2. What holiday results in the maximum sales for this department?
3. a) Generate linear and quadratic models for this data.
b) What is the marginal sales for this department using each model.
c) Which model do you feel best predicts future trends and explain your rational.
4. Based on the model selected, what type of seasonal adjustments, if any, would be
required to meet customer needs?
5. Some items were added or subtracted from the 2003-2004 dry goods department
data when compared with the data available for the previous year (2002-2003).
a) Use your best model for the 2003-2004 data set to predict sales for the next
four weeks. Provide chart and model backup for predictions.
b) Compute the percent rate of increase 2
1
y y /y
− 1 for the next four weeks using
results from part a). Provide appropriate backup material.
Week Sales in $
41 18000
42 16800
43 15200
44 15000
45 13600
46 16000
47 12600
48 14800
49 16800
50 14800
51 15200
52 16000
53 15600
54 15600
55 15000
56 15700
57 15800
58 13800
59 12800
60 14400
61 15800
62 16000
63 12400
64 16200
65 17000
66 18600
67 16000
68 18000
69 19600
70 18600
71 18450
72 18000
73 18200
74 18600
75 16000
76 15200
77 16800
78 15800
79 17600
80 15800
81 15600
82 14200
83 16600
84 16100
85 14100
86 14400
87 14500
88 16900
89 17000
90 16000
91 17800
The data is for weekly sales in the dry goods department at a Wal*Mart store in the Northeast. Peak values, I.e. spikes, usually occur at holiday periods. Week 1 is the first week of February 2003. To show continuity, week 1 of 2004 is represented as week 53 since week 53 represents the start of the 2004 fiscal year. Dollar values are adjusted in order to disguise true sales figures, but trends in the data are retained for analysis purposes. Note that for 2002-2003 data, Wal*Mart used 53 weeks for their fiscal year data.
1. To identify holiday periods or special events that cause spikes in the data, you can start by looking for recurring patterns in the data that coincide with known holidays or events. One way to do this is by visualizing the data on a graph and marking the dates of holidays or special events. You can also compare the sales data during holiday periods with non-holiday periods to see if there are significant differences. Additionally, you can analyze the data for any significant deviations from the expected sales trend.
2. To determine which holiday results in the maximum sales for this department, you can analyze the sales data for different holidays individually. Look for the holiday that consistently has the highest sales figures or exhibits a significant increase compared to other holidays. You can also compare the sales data for each holiday with the overall average sales to see which holiday has the highest percentage increase.
3. a) To generate linear and quadratic models for the data, you will need to perform regression analysis. Regression analysis uses mathematical techniques to find the best-fitting line or curve that represents the relationship between the independent variable (e.g., time) and the dependent variable (e.g., sales). The linear model will be of the form y = mx + b, and the quadratic model will be of the form y = ax^2 + bx + c, where y is the sales, x is the time, and m, b, a, b, and c are coefficients to be determined.
b) To find the marginal sales for this department using each model, you can calculate the derivative of the equations derived from the linear and quadratic models. The derivative will give you the rate of change of sales with respect to time. At any given point, the marginal sales will be the slope of the linear regression line or the instantaneous rate of change of the quadratic regression curve.
c) To determine which model best predicts future trends, you can consider several factors. Firstly, you can visually compare the fit of the linear and quadratic models to the actual data. A good fit would indicate that the model captures the underlying trend accurately. Secondly, you can use statistical measures such as the coefficient of determination (R-squared) or mean squared error (MSE) to assess the accuracy of each model. A model with a higher R-squared and lower MSE would be preferable. Lastly, you can consider the theoretical plausibility of each model in relation to the department's operations and sales patterns.
4. Based on the selected model, you can analyze the residuals (the differences between the model predictions and actual sales) to identify any systematic seasonal patterns. If the residuals show a consistent pattern, it suggests the presence of seasonality in the data. You can then apply seasonal adjustments to account for these patterns and better meet customer needs. Seasonal adjustments can involve adding or subtracting specific amounts from the predicted sales during certain periods to account for the seasonal effects.
5. a) To predict sales for the next four weeks using your best model for the 2003-2004 data set, you can extrapolate the trend line or curve. Use the equation of the selected model and substitute the appropriate time values to obtain the predicted sales for each week. You can then plot these predictions on a chart to visualize the projected sales.
b) To compute the percent rate of increase for the next four weeks, you can calculate the difference between the predicted sales for each week and divide it by the previous week's sales. Multiply the result by 100 to get the percentage increase. For example, ((predicted sales for week 2 - predicted sales for week 1) / predicted sales for week 1) * 100 will give you the percent rate of increase from week 1 to week 2. Repeat this calculation for all four weeks and provide appropriate backup material such as the predicted sales values and the calculation steps.