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To solve this exponential smoothing problem, we need to calculate the forecast for year 8 using the data provided and an alpha (smoothing constant) of 0.30.

### Steps to Solution:

1. **Calculate Initial Forecast (F4) for Year 4:**  
   The problem states to use the average demand for years 1 to 3 as the initial forecast.
   \[
   \text{Initial Forecast (F4)} = \frac{321 + 407 + 449}{3} = 392.33 \approx 392
   \]

2. **Apply Exponential Smoothing for Each Subsequent Year:**
   The exponential smoothing formula is:
   \[
   F_{t+1} = \alpha \times D_t + (1 - \alpha) \times F_t
   \]
   where:
   - \( F_{t+1} \) is the forecast for the next period,
   - \( D_t \) is the actual demand for the current period,
   - \( F_t \) is the forecast for the current period, and
   - \( \alpha = 0.30 \).

   **Calculate forecasts from year 4 to year 8.**

   - **Forecast for Year 5 (F5):**
     \[
     F_5 = 0.30 \times 507 + 0.70 \times 392 = 152.1 + 274.4 = 426.5 \approx 427
     \]

   - **Forecast for Year 6 (F6):**
     \[
     F_6 = 0.30 \times 499 + 0.70 \times 427 = 149.7 + 298.9 = 448.6 \approx 449
     \]

   - **Forecast for Year 7 (F7):**
     \[
     F_7 = 0.30 \times 551 + 0.70 \times 449 = 165.3 + 314.3 = 479.6 \approx 480
     \]

   - **Forecast for Year 8 (F8):**
     \[
     F_8 = 0.30 \times 461 + 0.70 \times 480 = 138.3 + 336 = 474.3 \approx 474
     \]

### Final Answer:
The forecast for the 8th year is approximately **474** cases of Merlot wine.

Would you like more details or have any questions about this solution?

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#### Related Questions:
1. How does the smoothing constant (\(\alpha\)) affect the forecast in exponential smoothing?
2. What would the forecast be for year 9 if the demand in year 8 was 427?
3. Can exponential smoothing be used for trend-based data, and if so, how?
4. How would the forecast change if a different initial forecast was chosen?
5. Why might a company prefer exponential smoothing over other forecasting methods?

#### Tip:
Exponential smoothing gives more weight to recent data, making it ideal for time series data without strong seasonal or trend patterns.

Using an exponential smoothing model with an alpha value of 0.30, estimate the smoothed value calculated as of the end of year 8. Use the average demand for years 1 through 3 as your initial forecast, and then smooth the forecast forward to year 8. Note: Round your intermediate calculations and final answer to the nearest whole number.

Calculate the forecast for year 8 using exponential smoothing with an alpha of 0.30, based on demand data from previous years. This problem provides step-by-step guidance on smoothing forecasts and rounding to the nearest whole number.

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