Answer:
Explanation:
Let's calculate the requested measures of forecast accuracy using the given time series data:
Week: 1 2 3 4 5 6
Value: 18 13 16 11 17 14
a. Mean Absolute Error (MAE):
\[ MAE = \frac{1}{n} \sum_{i=1}^{n} |Y_i - \hat{Y}_i| \]
where \( n \) is the number of observations, \( Y_i \) is the actual value, and \( \hat{Y}_i \) is the forecast value.
\[ MAE = \frac{1}{6} (|18-15| + |13-15| + |16-15| + |11-15| + |17-15| + |14-15|) \]
\[ MAE = \frac{1}{6} (3 + 2 + 1 + 4 + 2 + 1) = \frac{13}{6} \]
b. Mean Squared Error (MSE):
\[ MSE = \frac{1}{n} \sum_{i=1}^{n} (Y_i - \hat{Y}_i)^2 \]
\[ MSE = \frac{1}{6} ( (18-15)^2 + (13-15)^2 + (16-15)^2 + (11-15)^2 + (17-15)^2 + (14-15)^2 ) \]
\[ MSE = \frac{1}{6} (9 + 4 + 1 + 16 + 4 + 1) = \frac{35}{6} \]
c. Mean Absolute Percentage Error (MAPE):
\[ MAPE = \frac{1}{n} \sum_{i=1}^{n} \left| \frac{Y_i - \hat{Y}_i}{Y_i} \right| \times 100 \]
\[ MAPE = \frac{1}{6} \left( \left| \frac{18-15}{18} \right| + \left| \frac{13-15}{13} \right| + \left| \frac{16-15}{16} \right| + \left| \frac{11-15}{11} \right| + \left| \frac{17-15}{17} \right| + \left| \frac{14-15}{14} \right| \right) \times 100 \]
\[ MAPE \approx \frac{1}{6} (16.67 + 15.38 + 6.25 + 36.36 + 11.76 + 7.14) \]
\[ MAPE \approx \frac{93.56}{6} \approx 15.59 \]
d. Forecast for Week 7:
Since you're using the average of all historical data as a forecast, the forecast for Week 7 would be the average of the values in Weeks 1 to 6.
\[ \text{Forecast for Week 7} = \frac{18 + 13 + 16 + 11 + 17 + 14}{6} \]
\[ \text{Forecast for Week 7} = \frac{89}{6} \approx 14.83 \]
So, the forecast for Week 7 is approximately 14.83.