Fit a trend line by the method of semi-averages for the given data.
Year	2000	2001	2002	2003	2004	2005	2006
Production	105	 115	120	100	110	125	135

The method of semi-averages is a way to fit a trend line to a set of data by dividing the data into two equal parts, calculating the average of each part, and then drawing a line between these averages.

### Steps to Fit a Trend Line Using Semi-Averages:

1. **Divide the Data into Two Equal Parts**:
   Since there are 7 data points, we'll divide them into two parts. The first part will have 3 data points, and the second part will have 4 data points.

   - First part (Years 2000 to 2002): \(105, 115, 120\)
   - Second part (Years 2003 to 2006): \(100, 110, 125, 135\)

2. **Calculate the Average for Each Part**:
   - First part average:
     \[
     \text{Average}_1 = \frac{105 + 115 + 120}{3} = \frac{340}{3} = 113.33
     \]
   - Second part average:
     \[
     \text{Average}_2 = \frac{100 + 110 + 125 + 135}{4} = \frac{470}{4} = 117.5
     \]

3. **Determine the Midpoints of Each Part**:
   - Midpoint of the first part: \(\text{Year} = 2001\)
   - Midpoint of the second part: \(\text{Year} = 2004.5\)

4. **Plot These Averages**:
   - First part: Plot the point \((2001, 113.33)\)
   - Second part: Plot the point \((2004.5, 117.5)\)

5. **Fit the Trend Line**:
   - The slope (\(m\)) of the line between these two points is:
     \[
     m = \frac{117.5 - 113.33}{2004.5 - 2001} = \frac{4.17}{3.5} \approx 1.19
     \]
   - The equation of the line in the form \(y = mx + c\) can be found using the point-slope form. Using the point \((2001, 113.33)\):
     \[
     y - 113.33 = 1.19(x - 2001)
     \]
     Simplifying:
     \[
     y = 1.19x - 2387.81 + 113.33
     \]
     \[
     y = 1.19x - 2274.48
     \]

### Final Trend Line Equation:
\[
y = 1.19x - 2274.48
\]

This trend line can now be used to predict future values of production based on the year.

Would you like any further details or have any questions?

### Relative Questions:
1. How can we apply the method of least squares to find a trend line?
2. What are the limitations of the semi-averages method?
3. Can we use the semi-averages method for non-linear data?
4. How would the trend line change if the production data had an outlier?
5. What is the significance of the slope in the trend line equation?

### Tip:
When using the method of semi-averages, ensure the data is approximately linear. If not, consider alternative methods like moving averages or regression analysis.

Learn how to fit a trend line to production data from 2000 to 2006 using the method of semi-averages. This detailed solution includes step-by-step calculations for dividing the data, calculating averages, plotting points, and deriving the trend line equation. Suitable for advanced high school or college students studying statistics and data analysis.

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