https://file.aiquickdraw.com/mathbot/file/1725936284710qtr8j2hh.jpg

From the image, we have the following details regarding mileage over the past five years:

| Year  | Mileage |
|-------|---------|
| 1     | 3,100   |
| 2     | 4,000   |
| 3     | 3,500   |
| 4     | 3,750   |
| 5     | 3,700   |

### Part a) Using a 2-year moving average to forecast year 6:

The 2-year moving average forecast for year 6 is calculated by averaging the mileage from years 4 and 5:

\[
\text{Forecast for Year 6} = \frac{3,750 + 3,700}{2} = \frac{7,450}{2} = 3,725 \text{ miles}
\]

So, the forecast for year 6 is **3,725 miles**.

### Part b) MAD (Mean Absolute Deviation) for a 2-year moving average:

To calculate the MAD, we first need to compute the 2-year moving averages for years 3, 4, and 5. Then, we'll find the absolute deviations between the actual and forecasted values, and finally, calculate the average of these deviations.

1. **Forecast for Year 3** (based on years 1 and 2):  
   \[
   \frac{3,100 + 4,000}{2} = \frac{7,100}{2} = 3,550 \text{ miles}
   \]
   
   **Absolute Deviation for Year 3**:  
   \[
   |3,500 - 3,550| = 50 \text{ miles}
   \]

2. **Forecast for Year 4** (based on years 2 and 3):  
   \[
   \frac{4,000 + 3,500}{2} = \frac{7,500}{2} = 3,750 \text{ miles}
   \]
   
   **Absolute Deviation for Year 4**:  
   \[
   |3,750 - 3,750| = 0 \text{ miles}
   \]

3. **Forecast for Year 5** (based on years 3 and 4):  
   \[
   \frac{3,500 + 3,750}{2} = \frac{7,250}{2} = 3,625 \text{ miles}
   \]
   
   **Absolute Deviation for Year 5**:  
   \[
   |3,700 - 3,625| = 75 \text{ miles}
   \]

Now, calculate the MAD (Mean Absolute Deviation):
\[
\text{MAD} = \frac{50 + 0 + 75}{3} = \frac{125}{3} = 41.67 \text{ miles}
\]

Therefore, the MAD is approximately **41.7 miles**.

---

Would you like further details on any of the calculations or concepts?

### Related Questions:
1. How would the forecast change if a 3-year moving average was used?
2. Can we use exponential smoothing to forecast year 6? What would be the result?
3. How does the choice of the number of years for the moving average affect the accuracy of predictions?
4. What other measures besides MAD can be used to evaluate the accuracy of a forecast?
5. How does the data from previous years affect future predictions in moving average calculations?

### Tip:
MAD is a simple way to evaluate forecast accuracy, but in real-world applications, other methods like MSE (Mean Squared Error) or MAPE (Mean Absolute Percentage Error) are also frequently used for a more nuanced analysis.

The Carbondale Hospital is considering the purchase of a new ambulance. The decision will rest partly on the anticipated mileage to be driven next year. The miles driven during the past 5 years are as follows: 

Year: 1 - 5
Mileage: 3,100, 4,000, 3,500, 3,750, 3,700

Part a) Using a 2-year moving average, the forecast for year 6.
Part b) Calculate the MAD based on the 2-year moving average forecast.

Learn how to calculate the 2-year moving average and MAD (Mean Absolute Deviation) to forecast the mileage for Carbondale Hospital's new ambulance. This example uses mileage data from the past 5 years to predict future trends and assess forecast accuracy. Suitable for high school students.

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