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This problem involves finding a **linear regression equation** to model the given data set, which describes the value $ V(n) $ of an investment in dollars, $ n $ years after 1994. 

### The table of values provided is:

| $ n $ (years after 1994) | 1 | 3 | 7 | 12 | 14 | 19 |
|----------------------------|------|------|-------|-------|-------|-------|
| $ V(n) $ (value in $)     | 12270 | 11573.8 | 10890 | 10032.2 | 9280 | 7723.5 |

### To solve:
1. **Determine the linear regression equation**: This equation will have the form:
   \[
   V(n) = m \cdot n + b
   \]
   where:
   - $ m $ is the slope of the regression line (representing the rate of change in value per year).
   - $ b $ is the y-intercept, representing the value of the investment in the year 1994 (when $ n = 0 $).

2. **Calculate the slope $ m $**:
   The slope $ m $ is calculated as:
   \[
   m = \frac{\sum (n_i - \bar{n})(V_i - \bar{V})}{\sum (n_i - \bar{n})^2}
   \]
   where $ \bar{n} $ and $ \bar{V} $ are the means of $ n $ and $ V(n) $, respectively.

3. **Find the y-intercept $ b $**:
   Once $ m $ is known, use the equation $ V(n) = m \cdot n + b $ and substitute one of the known points (e.g., $ n = 1 $, $ V(n) = 12270 $) to find $ b $.

I will now calculate the slope $ m $ and the intercept $ b $ using the given data points to provide the full linear regression equation.The linear regression equation that models the given data is:

\[
V(n) = -236.97 \cdot n + 12506.64
\]

### Interpretation:
- The slope $ m = -236.97 $ indicates that the value of the investment decreases by approximately **$236.97 per year**.
- The y-intercept $ b = 12506.64 $ suggests that the value of the investment in the year 1994 (when $ n = 0 $) was approximately **$12,506.64**.

Let me know if you want further details or explanations.

Here are some related questions you can consider:
1. How do we interpret the slope in this context?
2. Can linear regression always be used to model investment trends?
3. What happens to the investment value 30 years after 1994?
4. How would the model change if the trend reversed (increasing instead of decreasing)?
5. What is the expected value of the investment in the year 2005?

**Tip**: When performing regression analysis, always check if a linear model is appropriate by analyzing the residuals.

Determine the linear regression equation that models the set of data for the value of an investment (in dollars) n years after 1994. Based on this regression model, determine the value of the investment in 1994 and the rate at which the investment is changing.

Learn how to determine the linear regression equation to model the investment value over time. Using provided data, this problem helps calculate the investment's value in 1994 and the rate of change. Key concepts include slope-intercept form and regression analysis. Suitable for students in Grades 10-12.

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