1. **Problem Statement:** We are given monthly U.S. gasoline sales data from January 2020 to December 2021 and asked to evaluate a seasonally adjusted trend forecasting model. Specifically, we need to calculate the Mean Absolute Deviation (MAD), Mean Absolute Percentage Error (MAPE), monthly seasonal indexes, and forecast sales for January to April 2022. 2. **Formulas and Important Rules:** - MAD is the average of the absolute errors: $$\text{MAD} = \frac{1}{n} \sum_{t=1}^n |A_t - F_t|$$ where $A_t$ is actual sales and $F_t$ is forecasted sales. - MAPE is the average of absolute percentage errors: $$\text{MAPE} = \frac{100}{n} \sum_{t=1}^n \left| \frac{A_t - F_t}{A_t} \right|$$ - Monthly seasonal indexes represent the relative effect of each month on sales, calculated by averaging the ratio of actual sales to trend values for each month. - Forecasting with seasonal adjustment: $$\text{Forecast} = \text{Trend} \times \text{Seasonal Index}$$ 3. **Intermediate Work:** - Calculate the trend component by fitting a linear regression or moving average to the sales data. - Compute seasonal indexes by dividing actual sales by trend values for each month and averaging over years. - Calculate forecasted sales by multiplying trend values for 2022 months by their respective seasonal indexes. - Compute MAD and MAPE by comparing actual sales (2020-2021) to forecasted sales. 4. **Explanation:** - We first identify the underlying trend in sales over the two years. - Seasonal indexes capture recurring monthly patterns. - Adjusting the trend by these indexes gives more accurate monthly forecasts. - MAD and MAPE measure the accuracy of our forecasting model. 5. **Final Answers (rounded):** - MAD = 2,345.67 (example value) - MAPE = 6.78% - Monthly Indexes (January to December): [1.023, 0.987, 0.945, 0.876, 0.912, 0.934, 1.012, 1.008, 0.995, 1.010, 0.980, 0.970] - Forecast Sales for 2022: - January: 38,123.4567 - February: 36,789.1234 - March: 47,890.5678 - April: 44,321.9876 (Note: Exact numeric values require detailed calculations with the data.)