To solve this problem, we will use the naïve method for forecasting, which means that the forecast for a given week is simply the value from the previous week.

Given dataTo solve this problem, we will use the naïve method for forecasting, which means that the forecast for a given week is simply the value from the previous week.

Given

• Week: 1, 2, 3, 4, 5, 6
• Value: 18, 13, 16, 11, 17, 14

Step 1: Generate Forecasts

• The forecast for week 2 is the value from week 1, which is 18.
• The forecast for week 3 is the value from week 2, which is 13.
• The forecast for week 4 is the value from week 3, which is 16.
• The forecast for week 5 is the value from week 4, which is 11.
• The forecast for week 6 is the value from week 5, which is 17.

So, the forecasts are:

• Forecast for week 2 = 18
• Forecast for week 3 = 13
• Forecast for week 4 = 16
• Forecast for week 5 = 11
• Forecast for week 6 = 17

Step 2: Compute Errors

The error for each week is calculated as the actual value minus the forecasted value:

• Error for week 2 = Actual (13) - Forecast (18) = -5
• Error for week 3 = Actual (16) - Forecast (13) = 3
• Error for week 4 = Actual (11) - Forecast (16) = -5
• Error for week 5 = Actual (17) - Forecast (11) = 6
• Error for week 6 = Actual (14) - Forecast (17) = -3

Step 3: Calculate Forecast Accuracy Measures

(a) Mean Absolute Error (MAE)

The mean absolute error is the average of the absolute errors: $\text{MAE} = \frac{|{-5}| + |{3}| + |{-5}| + |{6}| + |{-3}|}{5} = \frac{5 + 3 + 5 + 6 + 3}{5} = \frac{22}{5} = 4.4$ So, the MAE is 4.4.

(b) Mean Squared Error (MSE)

The mean squared error is the average of the squared errors: $\text{MSE} = \frac{(-5)^2 + (3)^2 + (-5)^2 + (6)^2 + (-3)^2}{5} = \frac{25 + 9 + 25 + 36 + 9}{5} = \frac{104}{5} = 20.8$ So, the MSE is 20.8.

(c) Mean Absolute Percentage Error (MAPE)

The mean absolute percentage error is the average of the absolute percentage errors: $\text{MAPE} = \frac{\frac{|{-5}|}{13} \times 100 + \frac{|{3}|}{16} \times 100 + \frac{|{-5}|}{11} \times 100 + \frac{|{6}|}{17} \times 100 + \frac{|{-3}|}{14} \times 100}{5}$ Calculating each term: $\frac{|{-5}|}{13} \times 100 = 38.46\%, \quad \frac{|{3}|}{16} \times 100 = 18.75\%, \quad \frac{|{-5}|}{11} \times 100 = 45.45\%$ $\frac{|{6}|}{17} \times 100 = 35.29\%, \quad \frac{|{-3}|}{14} \times 100 = 21.43\%$ Now, calculate the average: $\text{MAPE} = \frac{38.46 + 18.75 + 45.45 + 35.29 + 21.43}{5} = \frac{159.38}{5} = 31.88\%$ So, the MAPE is 31.88%.

(d) Forecast for Week 7

Using the naïve method, the forecast for week 7 is the actual value from week 6, which is 14.

Summary:

(a) MAE = 4.4
(b) MSE = 20.8
(c) MAPE = 31.88%
(d) Forecast for week 7 = 14

Do you have any questions, or would you like more details on any part of the solution?

Here are 5 related questions to expand your understanding:

1. How does the naïve forecasting method compare to other methods like moving average or exponential smoothing?
2. Why is MAPE useful when comparing forecast accuracy across different data series?
3. What are the limitations of using the mean squared error (MSE) as a measure of forecast accuracy?
4. How would forecast accuracy change if we used a more advanced forecasting model like ARIMA?
5. What impact does a larger error in one period have on the different accuracy measures?

Tip: When interpreting MAPE, remember that it gives you a percentage error, which can help compare different time series even if they have different scales.