Consider the following time series data.

Week	1	2	3	4	5	6
Value	18	13	16	11	17	14
Using the naïve method (most recent value) as the forecast for the next week, compute the following measures of forecast accuracy.

(a)	Mean absolute error
If required, round your answer to one decimal place.
 	 
(b)	Mean squared error
If required, round your answer to one decimal place.
 	 
(c)	Mean absolute percentage error
If required, round your intermediate calculations and final answer to two decimal places.
 	 
(d)	What is the forecast for week 7?


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.

Consider the following time series data.

Week 1 2 3 4 5 6
Value 18 13 16 11 17 14
Using the naïve method (most recent value) as the forecast for the next week, compute the following measures of forecast accuracy.

(a) Mean absolute error
(b) Mean squared error
(c) Mean absolute percentage error
(d) What is the forecast for week 7?

Learn how to calculate forecast accuracy using the naïve method for a time series dataset. This example covers Mean Absolute Error (MAE), Mean Squared Error (MSE), and Mean Absolute Percentage Error (MAPE), including step-by-step instructions. Suitable for students studying statistics or business analytics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation