To calculate the simple exponential smoothing forecast for the 4th week using the time series data provided and a smoothing constant (α) of 0.4, we will follow these steps:
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Initialization: We need an initial forecast for week 1. A common practice is to set this equal to the first observed value:
- \( F_1 = 6.00 \)
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Compute forecasts for subsequent weeks: The formula for exponential smoothing is: \[ F_t = \alpha \cdot Y_{t-1} + (1 - \alpha) \cdot F_{t-1} \] where:
- \( F_t \) = forecast for week \( t \)
- \( Y_{t-1} \) = actual value from the previous week
- \( F_{t-1} \) = forecast from the previous week
- \( \alpha \) = smoothing constant (0.4 in this case)
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Calculating the forecast for week 2:
- Actual value \( Y_1 = 6.00 \)
- Forecast \( F_1 = 6.00 \)
\[ F_2 = 0.4 \cdot 6.00 + (1 - 0.4) \cdot 6.00 \] \[ F_2 = 0.4 \cdot 6.00 + 0.6 \cdot 6.00 = 2.4 + 3.6 = 6.00 \]
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Calculating the forecast for week 3:
- Actual value \( Y_2 = 4.00 \)
\[ F_3 = 0.4 \cdot 4.00 + (1 - 0.4) \cdot 6.00 \] \[ F_3 = 0.4 \cdot 4.00 + 0.6 \cdot 6.00 = 1.6 + 3.6 = 5.20 \]
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Calculating the forecast for week 4:
- Actual value \( Y_3 = 4.00 \)
\[ F_4 = 0.4 \cdot 4.00 + (1 - 0.4) \cdot 5.20 \] \[ F_4 = 0.4 \cdot 4.00 + 0.6 \cdot 5.20 = 1.6 + 3.12 = 4.72 \]
Therefore, the simple exponential smoothing forecast for the 4th week is 4.72.