The document appears to focus on forecasting sales data with time series analysis, specifically using exponential smoothing. Here’s how we can approach solving each part:

Part A: Time Series Plot

1. Plot the Sales Data: Using the data provided for each month, we can create a line chart to represent the time series. Here’s the data as provided:
   • Months: 1 through 12
   • Sales: [100, 140, 125, 110, 95, 125, 150, 135, 100, 75, 95, 110]

Part B: Exponential Smoothing with Alpha = 0.2

1. Initial Forecast: The forecast for the first month is usually taken as the actual sales for that month, i.e., 100.
2. Formula: The exponential smoothing formula is: $F_{t} = \alpha Y_{t-1} + (1 - \alpha) F_{t-1}$ where:
   • $F_{t}$ is the forecast for month $t$,
   • $Y_{t-1}$ is the actual value for the previous month,
   • $F_{t-1}$ is the forecast for the previous month, and
   • $\alpha = 0.2$.
3. Calculations: Using this formula, calculate the forecast for each month and compute the errors by subtracting the forecast from the actual values.

Part C: Exponential Smoothing with Alpha = 0.5

1. Repeat the Forecasting Process with $\alpha = 0.5$, following the same steps as Part B.
2. Calculate Errors for each month again.

Mean Squared Error (MSE)

1. Compute MSE for each smoothing constant ($\alpha = 0.2$ and $\alpha = 0.5$) by averaging the squared forecast errors over the period.

Would you like detailed calculations for each part? Here’s a quick breakdown of what we can explore further:

1. How to set up and calculate the forecasts manually.
2. Generating the time series plot for visual representation.
3. Explanation of how the smoothing constant affects the forecast sensitivity.
4. Steps to compute MSE for comparison between alpha values.
5. Interpretation of the results and suggestions for the best smoothing constant.

Tip:

When using exponential smoothing, lower alpha values yield smoother forecasts but react slower to changes, while higher alpha values make forecasts more responsive to recent data changes.