Question 1171314: Time Series and Non-Parametric Tests
1. Perform a Trend and Seasonal analysis with the following data. Include the trend formula
and the multiplicative indices for the seasons. Using the model forecast sales for the next
year (all 4 seasons of year 4)
Year Season Sales
1 sp 140
1 su 50
1 w 130
1 f 520
2 sp 200
2 su 110
2 w 190
2 f 550
3 sp 260
3 su 220
3 w 210
3 f 570
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Absolutely! Let's break down this time series analysis step by step.
**1. Trend Analysis**
First, we'll find the trend line. To do this, we need to assign numerical values to the periods. We'll use the following numbering:
* Year 1, Spring (sp): 1
* Year 1, Summer (su): 2
* Year 1, Winter (w): 3
* Year 1, Fall (f): 4
* Year 2, Spring (sp): 5
* ... and so on
Let's create a table with the period numbers (t) and the sales values (y):
| t | Year | Season | Sales (y) |
|---|------|--------|-----------|
| 1 | 1 | sp | 140 |
| 2 | 1 | su | 50 |
| 3 | 1 | w | 130 |
| 4 | 1 | f | 520 |
| 5 | 2 | sp | 200 |
| 6 | 2 | su | 110 |
| 7 | 2 | w | 190 |
| 8 | 2 | f | 550 |
| 9 | 3 | sp | 260 |
| 10 | 3 | su | 220 |
| 11 | 3 | w | 210 |
| 12 | 3 | f | 570 |
Now, we calculate the trend line using linear regression.
* **Calculate the means:**
* $\bar{t} = \frac{\sum t}{n} = \frac{1+2+3+...+12}{12} = \frac{78}{12} = 6.5$
* $\bar{y} = \frac{\sum y}{n} = \frac{140+50+130+520+200+110+190+550+260+220+210+570}{12} = \frac{2950}{12} \approx 245.83$
* **Calculate the sums:**
* $\sum t^2 = 1^2 + 2^2 + ... + 12^2 = 650$
* $\sum ty = (1)(140) + (2)(50) + ... + (12)(570) = 22170$
* **Calculate the slope (b):**
* $b = \frac{n \sum ty - (\sum t)(\sum y)}{n \sum t^2 - (\sum t)^2}$
* $b = \frac{(12)(22170) - (78)(2950)}{(12)(650) - (78)^2}$
* $b = \frac{266040 - 230100}{7800 - 6084} = \frac{35940}{1716} \approx 20.94$
* **Calculate the intercept (a):**
* $a = \bar{y} - b\bar{t}$
* $a = 245.83 - (20.94)(6.5) \approx 245.83 - 136.11 \approx 109.72$
* **Trend Formula:**
* $Trend = a + bt$
* $Trend = 109.72 + 20.94t$
**2. Seasonal Indices**
* **Calculate the trend values for each period:**
* Use the trend formula to calculate the trend value for each period (t).
| t | Year | Season | Sales (y) | Trend |
|---|------|--------|-----------|-------|
| 1 | 1 | sp | 140 | 130.66 |
| 2 | 1 | su | 50 | 151.60 |
| 3 | 1 | w | 130 | 172.54 |
| 4 | 1 | f | 520 | 193.48 |
| 5 | 2 | sp | 200 | 214.42 |
| 6 | 2 | su | 110 | 235.36 |
| 7 | 2 | w | 190 | 256.30 |
| 8 | 2 | f | 550 | 277.24 |
| 9 | 3 | sp | 260 | 298.18 |
| 10 | 3 | su | 220 | 319.12 |
| 11 | 3 | w | 210 | 340.06 |
| 12 | 3 | f | 570 | 361.00 |
* **Calculate the seasonal ratios:**
* Divide the actual sales (y) by the trend value for each period.
| t | Year | Season | Sales (y) | Trend | Ratio (y/Trend) |
|---|------|--------|-----------|-------|-----------------|
| 1 | 1 | sp | 140 | 130.66 | 1.0715 |
| 2 | 1 | su | 50 | 151.60 | 0.3298 |
| 3 | 1 | w | 130 | 172.54 | 0.7534 |
| 4 | 1 | f | 520 | 193.48 | 2.6876 |
| 5 | 2 | sp | 200 | 214.42 | 0.9328 |
| 6 | 2 | su | 110 | 235.36 | 0.4674 |
| 7 | 2 | w | 190 | 256.30 | 0.7413 |
| 8 | 2 | f | 550 | 277.24 | 1.9839 |
| 9 | 3 | sp | 260 | 298.18 | 0.8720 |
| 10 | 3 | su | 220 | 319.12 | 0.6894 |
| 11 | 3 | w | 210 | 340.06 | 0.6175 |
| 12 | 3 | f | 570 | 361.00 | 1.5789 |
* **Calculate the average seasonal ratios:**
* Average the ratios for each season.
* Spring: (1.0715 + 0.9328 + 0.8720) / 3 = 0.9588
* Summer: (0.3298 + 0.4674 + 0.6894) / 3 = 0.4955
* Winter: (0.7534 + 0.7413 + 0.6175) / 3 = 0.7041
* Fall: (2.6876 + 1.9839 + 1.5789) / 3 = 2.0835
* **Adjust for a mean of 1:**
* Calculate the average of the average seasonal ratios: (0.9588 + 0.4955 + 0.7041 + 2.0835) / 4 = 1.060475
* Divide each average seasonal ratio by 1.060475.
* Spring:
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