Two independent methods of forecasting based on judgment and experience have been prepared each month for the past 10 months. The forecasts and actual sales are as follows:

Month	Sales	Forecast 1	Forecast 2
1	835	800	810
2	835	825	805
3	830	830	790
4	805	815	800
5	780	790	800
6	850	770	806
7	800	780	765
8	840	815	805
9	835	780	835
10	805	820	795
Compute the MSE and MAD for each forecast.

Note: Round your answers to 2 decimal places.

Compute MAPE for each forecast.

Note: Round your intermediate calculations to 5 decimal places and final answers to 4 decimal places.

Prepare a naive forecast for periods 2 through 11 using the given sales data. Compute each of the following;

MSE,
MAD,
Tracking signal at month 10, and
2s control limits.
Note: Negative values should be indicated by a minus sign. Round your answers to 2 decimal places.

Let's break this task down into several parts, calculating each metric step by step for both forecast methods and then proceeding with the naive forecast.

### 1. Mean Squared Error (MSE)
The formula for MSE is:
\[
\text{MSE} = \frac{1}{n} \sum_{i=1}^{n} (F_i - A_i)^2
\]
where \(F_i\) is the forecast value, \(A_i\) is the actual value, and \(n\) is the number of periods.

### 2. Mean Absolute Deviation (MAD)
The formula for MAD is:
\[
\text{MAD} = \frac{1}{n} \sum_{i=1}^{n} |F_i - A_i|
\]

### 3. Mean Absolute Percentage Error (MAPE)
The formula for MAPE is:
\[
\text{MAPE} = \frac{100}{n} \sum_{i=1}^{n} \left| \frac{A_i - F_i}{A_i} \right|
\]

We'll compute MSE, MAD, and MAPE for both Forecast 1 and Forecast 2 based on the given sales data.

### 4. Naive Forecast
For a naive forecast, each forecast for period \(t\) is simply the actual sales from the previous period, i.e., \( \hat{S_t} = S_{t-1} \).

We'll use this naive forecast to compute the following metrics:

- **MSE**, **MAD**, and **Tracking Signal** at month 10.
- **2s Control Limits**, which involve standard deviations from the mean.

I'll calculate these metrics step by step.Here are the calculated metrics based on the given dataHere are the calculated metrics based on the given 

### Forecast 1:
- **MSE:** 1220.00
- **MAD:** 26.00
- **MAPE:** 3.1252%

### Forecast 2:
- **MSE:** 803.60
- **MAD:** 24.40
- **MAPE:** 2.9552%

### Naive Forecast (for periods 2 through 11):
- **MSE:** 1244.44
- **MAD:** 27.78
- **Tracking Signal (at month 10):** -1.08
- **2s Control Limits:** 70.24

Would you like further details on any of the calculations or concepts?

Here are 5 questions for further exploration:
1. How does the MAPE differ between Forecast 1 and Forecast 2, and why might that be important?
2. What would happen if one of the forecasts had a large outlier in error? How would that impact MSE vs MAD?
3. What is the significance of the tracking signal and how is it used in real-time forecasting?
4. How are 2s control limits useful in evaluating forecast accuracy?
5. Why is it important to use different error metrics like MSE, MAD, and MAPE?

**Tip:** MAPE is useful when comparing forecasting performance across different scales, as it expresses the error as a percentage of the actual value.

Learn how to compute MSE, MAD, and MAPE for two forecast methods based on judgment and experience, using sales data from the last 10 months. This problem also involves calculating a naive forecast for future periods, tracking signals, and control limits. Includes detailed step-by-step solutions with error metrics rounded to 2 decimal places.

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