To calculate the simple exponential smoothing (SES) forecast for the number of sports equipment that will be sold in 2022 with a smoothing factor $\alpha = 0.3$, we need to use the formula:

$F_{t+1} = \alpha \cdot A_t + (1 - \alpha) \cdot F_t$

Where:

• $F_{t+1}$ is the forecast for the next period (2022 in our case),
• $A_t$ is the actual value for the current period,
• $F_t$ is the forecast for the current period,
• $\alpha = 0.3$ is the smoothing factor.

Given that the forecast for 2015 is assumed to be the actual value of 2015 ($F_{2015} = 745$), we can proceed year by year to calculate the forecast up to 2022.

Let's go through the calculations step by step.

1. Forecast for 2016:

$F_{2016} = \alpha \cdot A_{2015} + (1 - \alpha) \cdot F_{2015}$ $F_{2016} = 0.3 \cdot 745 + 0.7 \cdot 745 = 745$

2. Forecast for 2017:

$F_{2017} = \alpha \cdot A_{2016} + (1 - \alpha) \cdot F_{2016}$ $F_{2017} = 0.3 \cdot 742 + 0.7 \cdot 745 = 743.10$

3. Forecast for 2018:

$F_{2018} = \alpha \cdot A_{2017} + (1 - \alpha) \cdot F_{2017}$ $F_{2018} = 0.3 \cdot 713 + 0.7 \cdot 743.10 = 733.17$

4. Forecast for 2019:

$F_{2019} = \alpha \cdot A_{2018} + (1 - \alpha) \cdot F_{2018}$ $F_{2019} = 0.3 \cdot 689 + 0.7 \cdot 733.17 = 719.42$

5. Forecast for 2020:

$F_{2020} = \alpha \cdot A_{2019} + (1 - \alpha) \cdot F_{2019}$ $F_{2020} = 0.3 \cdot 510 + 0.7 \cdot 719.42 = 655.59$

6. Forecast for 2021:

$F_{2021} = \alpha \cdot A_{2020} + (1 - \alpha) \cdot F_{2020}$ $F_{2021} = 0.3 \cdot 755 + 0.7 \cdot 655.59 = 685.41$

7. Forecast for 2022:

$F_{2022} = \alpha \cdot A_{2021} + (1 - \alpha) \cdot F_{2021}$ $F_{2022} = 0.3 \cdot 525 + 0.7 \cdot 685.41 = 637.79$

Final Result:

The simple exponential smoothing forecast for the number of sports equipment sold in 2022 is 637.79.

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Do you need further details or have any questions?

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Here are 5 related questions to explore:

1. How does the smoothing factor $\alpha$ affect the forecast in exponential smoothing?
2. What would the forecast be if $\alpha = 0.5$ instead?
3. How would a trend or seasonality affect simple exponential smoothing?
4. Can exponential smoothing be used for short-term vs long-term forecasts, and what are the limitations?
5. How does exponential smoothing compare to other forecasting methods like moving averages?

Tip: In exponential smoothing, lower $\alpha$ values give more weight to past forecasts, making the model slower to react to recent changes.