To analyze this data and calculate the required elements, we will follow the steps below for a thorough understanding.
### Given Data:
We have the following data of annual profits for certain years:
```
Years: 2001, 2002, 2003, 2004, 2005, 2006, 2007
Profits (in thousands): 60, 12, 15, 65, 80, 85, 97
```
### 1. Develop the Linear Equation
Using the method of least squares, we aim to find a linear relationship between the year and the profit of the form:
[tex]\[ \text{Profit} = \text{slope} \times \text{Year} + \text{intercept} \][/tex]
Given the results:
- Slope ([tex]\(m\)[/tex]): [tex]\(11.5\)[/tex]
- Intercept ([tex]\(c\)[/tex]): [tex]\(-22986.85714285714\)[/tex]
Thus, the linear equation describing the trend is:
[tex]\[ \text{Profit} = 11.5 \times \text{Year} - 22986.85714285714 \][/tex]
### 2. Calculate and Tabulate the Trend Values
Using the linear equation, we can compute the trend values for each year.
For each year (x), the profit is calculated as:
[tex]\[ \text{Profit} = 11.5 \times \text{Year} - 22986.85714285714 \][/tex]
The trend values for the years 2001 to 2007 are as follows:
- For 2001: [tex]\(11.5 \times 2001 - 22986.85714285714 = 24.64285714\)[/tex]
- For 2002: [tex]\(11.5 \times 2002 - 22986.85714285714 = 36.14285714\)[/tex]
- For 2003: [tex]\(11.5 \times 2003 - 22986.85714285714 = 47.64285714\)[/tex]
- For 2004: [tex]\(11.5 \times 2004 - 22986.85714285714 = 59.14285714\)[/tex]
- For 2005: [tex]\(11.5 \times 2005 - 22986.85714285714 = 70.64285714\)[/tex]
- For 2006: [tex]\(11.5 \times 2006 - 22986.85714285714 = 82.14285714\)[/tex]
- For 2007: [tex]\(11.5 \times 2007 - 22986.85714285714 = 93.64285714\)[/tex]
### 3. Calculate the Short-term Fluctuations
Short-term fluctuations are calculated by subtracting the trend value from the actual profit value for each corresponding year.
### 4. Monthly Increase in Profit
The monthly increase in profit can be derived from the annual slope:
[tex]\[ \text{Monthly Increase} = \frac{\text{Annual Slope}}{12} = \frac{11.5}{12} = 0.9583333333333334 \, \text{(thousands per month)} \][/tex]
### 5. Estimate the Profit for 2014
Using the linear equation, we can estimate the profit for the year 2014:
[tex]\[ \text{Estimated Profit for 2014} = 11.5 \times 2014 - 22986.85714285714 = 174.1428571428587 \, \text{(thousands)} \][/tex]
### Summary:
- Trend Equation: [tex]\( \text{Profit} = 11.5 \times \text{Year} - 22986.85714285714 \)[/tex]
- Trend Values:
- 2001: 24.64285714
- 2002: 36.14285714
- 2003: 47.64285714
- 2004: 59.14285714
- 2005: 70.64285714
- 2006: 82.14285714
- 2007: 93.64285714
- Monthly Increase in Profit: [tex]\(0.9583333333333334\)[/tex] thousands
- Estimated Profit for 2014: [tex]\(174.1428571428587\)[/tex] thousands
