To compute the forecast for the 3rd week using Simple Exponential Smoothing (SES), we can use the following formula:
\[ F_t = \alpha \times Y_{t-1} + (1 - \alpha) \times F_{t-1} \]
Where:
- \(F_t\) is the forecast for the current period,
- \(\alpha\) is the smoothing constant,
- \(Y_{t-1}\) is the actual value from the previous period,
- \(F_{t-1}\) is the forecast for the previous period.
Steps to calculate the forecasts for each week:
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Initial Forecast for Week 1: In SES, a common method for the initial forecast \(F_1\) is to use the first actual observation, \(Y_1\). Thus, we set: \[ F_1 = Y_1 = 10.00 \]
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Forecast for Week 2: Using the formula: \[ F_2 = \alpha \times Y_1 + (1 - \alpha) \times F_1 \] Substituting the values: \[ F_2 = 0.3 \times 10 + 0.7 \times 10 = 10.00 \]
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Forecast for Week 3: Using the forecast from Week 2: \[ F_3 = \alpha \times Y_2 + (1 - \alpha) \times F_2 \] Substituting the values: \[ F_3 = 0.3 \times 21 + 0.7 \times 10 = 6.3 + 7 = 13.3 \]
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Forecast for Week 4: (if needed) \[ F_4 = \alpha \times Y_3 + (1 - \alpha) \times F_3 \] Substituting the values: \[ F_4 = 0.3 \times 20 + 0.7 \times 13.3 = 6 + 9.31 = 15.31 \]
Final Forecast for Week 3
Thus, the forecast for the 3rd week is:
\[ \boxed{13.3} \]