Question 1171314
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: