(a) Compute MSE using the most recent value as the forecast for the next period.

The forecast for each month is the value of the previous month. Calculate the Mean Squared Error (MSE) using:

$\text{MSE} = \frac{1}{n}\sum_{i=1}^{n} (y_i - \hat{y}_i)^2$

Where $y_i$ is the actual value, and $\hat{y}_i$ is the forecasted value.

• Month 2 forecast = Value from Month 1 = 31
• Month 3 forecast = Value from Month 2 = 20
• Month 4 forecast = Value from Month 3 = 27
• Month 5 forecast = Value from Month 4 = 19
• Month 6 forecast = Value from Month 5 = 26
• Month 7 forecast = Value from Month 6 = 30

Calculate squared errors:

$\text{Squared Error for Month 2} = (20 - 31)^2 = 121$ $\text{Squared Error for Month 3} = (27 - 20)^2 = 49$ $\text{Squared Error for Month 4} = (19 - 27)^2 = 64$ $\text{Squared Error for Month 5} = (26 - 19)^2 = 49$ $\text{Squared Error for Month 6} = (30 - 26)^2 = 16$ $\text{Squared Error for Month 7} = (22 - 30)^2 = 64$

Sum of squared errors: $121 + 49 + 64 + 49 + 16 + 64 = 363$

Number of periods ($n$) = 6.

$\text{MSE} = \frac{363}{6} = 60.5$

MSE (Most Recent Value Forecast) = 60.5

Forecast for Month 8:

The forecast is the value from Month 7 = 22.

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(b) Compute MSE using the average of all available data as the forecast.

The forecast for each month is the average of all data available up to that point.

1. Month 2 forecast = Average of Month 1 = $31$
2. Month 3 forecast = Average of Months 1 & 2 = $\frac{31 + 20}{2} = 25.5$
3. Month 4 forecast = Average of Months 1–3 = $\frac{31 + 20 + 27}{3} = 26$
4. Month 5 forecast = Average of Months 1–4 = $\frac{31 + 20 + 27 + 19}{4} = 24.25$
5. Month 6 forecast = Average of Months 1–5 = $\frac{31 + 20 + 27 + 19 + 26}{5} = 24.6$
6. Month 7 forecast = Average of Months 1–6 = $\frac{31 + 20 + 27 + 19 + 26 + 30}{6} = 25.5$

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Calculate squared errors:

$\text{Squared Error for Month 2} = (20 - 31)^2 = 121$ $\text{Squared Error for Month 3} = (27 - 25.5)^2 = 2.25$ $\text{Squared Error for Month 4} = (19 - 26)^2 = 49$ $\text{Squared Error for Month 5} = (26 - 24.25)^2 = 3.06$ $\text{Squared Error for Month 6} = (30 - 24.6)^2 = 29.16$ $\text{Squared Error for Month 7} = (22 - 25.5)^2 = 12.25$

Sum of squared errors: $121 + 2.25 + 49 + 3.06 + 29.16 + 12.25 = 216.72$

Number of periods ($n$) = 6.

$\text{MSE} = \frac{216.72}{6} = 36.12$

MSE (Average Forecast) = 36.12

Forecast for Month 8:

The forecast is the average of all seven months: $\text{Average} = \frac{31 + 20 + 27 + 19 + 26 + 30 + 22}{7} = \frac{175}{7} = 25$

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(c) Which method provides the better forecast?

Conclusion: The average forecast has a lower MSE (36.12) than the most recent value forecast (60.5), so the average of all previous values provides a better forecast.

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Summary of Results:

1. MSE (Most Recent Value) = 60.5
2. Forecast for Month 8 (Most Recent Value) = 22
3. MSE (Average) = 36.12
4. Forecast for Month 8 (Average) = 25
5. Better Forecast Method: Average (smaller MSE).

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Would you like detailed steps for any part?

Questions to Explore:

1. How are different forecasting methods useful in real-world scenarios?
2. Why does MSE emphasize large errors more than small errors?
3. What other accuracy metrics could we use instead of MSE?
4. Could we use weighted averages for better forecasting? How?
5. How would adding more data points change the forecast accuracy?

Tip: Always compare MSE values when choosing the best forecasting method!