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To solve this problem, we will apply linear regression to the data provided in the table. The table gives values of an investment $ V(n) $ in dollars for different years $ n $ after 1992. 

We need to find the linear regression equation of the form:

\[
V(n) = mn + b
\]

Where:
- $ V(n) $ is the value of the investment at year $ n $,
- $ n $ is the number of years after 1992,
- $ m $ is the slope (the rate of change of the value per year),
- $ b $ is the y-intercept (the value of the investment in the year 1992 when $ n = 0 $).

### Step 1: Extract the data points
From the table:
- $ n = 1 $, $ V(1) = 19033 $
- $ n = 3 $, $ V(3) = 17933.02 $
- $ n = 7 $, $ V(7) = 16831 $
- $ n = 12 $, $ V(12) = 15445.88 $
- $ n = 14 $, $ V(14) = 14262 $
- $ n = 19 $, $ V(19) = 11805.65 $

### Step 2: Calculate the slope $ m $
The slope of the line is calculated using the formula:

\[
m = \frac{\sum (n - \bar{n})(V(n) - \bar{V(n)})}{\sum (n - \bar{n})^2}
\]

Where $ \bar{n} $ and $ \bar{V(n)} $ are the means of $ n $ and $ V(n) $, respectively.

### Step 3: Calculate the intercept $ b $
Once the slope $ m $ is known, the intercept $ b $ can be found using:

\[
b = \bar{V(n)} - m \bar{n}
\]

### Step 4: Use the linear regression equation to find:
- The value of the investment in 1992 (which is when $ n = 0 $, so $ b $).
- The rate of change of the value of the investment (which is the slope $ m $).

Let me calculate these values for you.The linear regression equation that models the set of data is:

\[
V(n) = -377.49n + 19408.30
\]

### Based on this regression model:
1. The value of the investment in the year 1992 was **\$19,408.30**.
2. The value of the investment is decreasing at a rate of **\$377.49** per year.

Would you like more details or further explanation on any part of the process?

Here are some related questions you can explore:
1. How do you calculate the slope in a linear regression?
2. Why does the investment value decrease over time in this model?
3. What would be the investment value after 25 years according to the model?
4. Can this model be used to predict future values of the investment? If so, how accurate might it be?
5. How does linear regression handle non-linear data?

**Tip:** When applying linear regression, the correlation between your variables can indicate how well the model will predict future data points.

Determine the linear regression equation that models the set of data given in the table of investment values, and use it to answer related questions about the investment's initial value and rate of change over time.

Learn how to determine the linear regression equation for modeling the value of an investment over time using data from a table. Includes step-by-step calculation of the slope and intercept, and how to use the model to find the investment's initial value and rate of change.

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