To forecast the value for May using the exponential smoothing method with $\alpha = 0.5$, we use the formula:

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

Where:

• $F_t$ is the forecast for time $t$.
• $A_{t-1}$ is the actual value at time $t-1$.
• $F_{t-1}$ is the forecast for time $t-1$.

For the initial forecast, $F_1$, we can use the actual value for January since no previous forecast is available. Let's start with $F_{January} = A_{January} = 21$.

Now we can calculate the forecasts for the following months step-by-step:

1. February forecast (F_{February}): $F_{February} = \alpha \cdot A_{January} + (1 - \alpha) \cdot F_{January}$ $F_{February} = 0.5 \cdot 21 + 0.5 \cdot 21$ $F_{February} = 10.5 + 10.5$ $F_{February} = 21$

2. March forecast (F_{March}): $F_{March} = \alpha \cdot A_{February} + (1 - \alpha) \cdot F_{February}$ $F_{March} = 0.5 \cdot 17 + 0.5 \cdot 21$ $F_{March} = 8.5 + 10.5$ $F_{March} = 19$

3. April forecast (F_{April}): $F_{April} = \alpha \cdot A_{March} + (1 - \alpha) \cdot F_{March}$ $F_{April} = 0.5 \cdot 19 + 0.5 \cdot 19$ $F_{April} = 9.5 + 9.5$ $F_{April} = 19$

4. May forecast (F_{May}): $F_{May} = \alpha \cdot A_{April} + (1 - \alpha) \cdot F_{April}$ $F_{May} = 0.5 \cdot 24 + 0.5 \cdot 19$ $F_{May} = 12 + 9.5$ $F_{May} = 21.5$

Thus, the forecast for May using the exponential smoothing method with $\alpha = 0.5$ is 21.5.

Answer:

21.5

Would you like more details or have any questions?

Here are some related questions:

1. How would the forecast change if $\alpha$ were 0.3 instead of 0.5?
2. What is exponential smoothing, and why is it used in time series forecasting?
3. Can you explain how the value of $\alpha$ affects the forecast in exponential smoothing?
4. What are some other methods of time series forecasting?
5. How can we measure the accuracy of our forecast?
6. What is the difference between simple exponential smoothing and double exponential smoothing?
7. Can you demonstrate exponential smoothing with a different set of data?
8. How would you apply exponential smoothing to forecast seasonal data?

Tip:

When choosing the smoothing parameter $\alpha$ in exponential smoothing, a higher $\alpha$ gives more weight to recent observations, making the forecast more responsive to changes in the data. Conversely, a lower $\alpha$ makes the forecast smoother and less responsive to recent changes.