Math Problem Statement
The table below shows the value, V, of an investment (in dollars) n years after 1989.
n
1
3
7
12
14
19
V
20431
20671.14
22417
24794.16
24734
25069.55
Determine the linear regression equation that models the set of data above, and use this equation to answer the questions below. Round to the nearest hundredth as needed.
Based on this regression model, the value of this investment was $in the year 1989.
Based on the regression model, the value of this investment is
at a rate of $per year.
Solution
To determine the linear regression equation for the investment data, we'll begin by establishing the format of the equation:
Where:
- is the value of the investment,
- is the number of years after 1989,
- is the slope of the line (the rate of change per year),
- is the y-intercept (the value of the investment in the year 1989).
Step 1: Data Preparation
Here is the data provided:
\hline n \text{ (years after 1989)} & V \text{ (Value of investment in dollars)} \\ \hline 1 & 20431 \\ 3 & 20671.14 \\ 7 & 22417 \\ 12 & 24794.16 \\ 14 & 24734 \\ 19 & 25069.55 \\ \hline \end{array}$$ ### Step 2: Calculating the Slope $$m$$ and the Intercept $$b$$ To calculate the linear regression equation, we use the following formulas for the slope $$m$$ and the intercept $$b$$: $$m = \frac{n \sum V - \sum n \sum V}{n \sum n^2 - (\sum n)^2}$$ $$b = \frac{\sum V - m \sum n}{n}$$ Where $$n$$ is the number of data points (6 in this case), and the summation terms represent: - $$\sum n$$ = sum of all $$n$$-values, - $$\sum V$$ = sum of all $$V$$-values, - $$\sum n^2$$ = sum of squares of the $$n$$-values, - $$\sum nV$$ = sum of the product of corresponding $$n$$ and $$V$$-values. Let me calculate this now. numpy as np # Given data n_values = np.array([1, 3, 7, 12, 14, 19]) V_values = np.array([20431, 20671.14, 22417, 24794.16, 24734, 25069.55]) # Calculating the sums required for the formulas sum_n = np.sum(n_values) sum_V = np.sum(V_values) sum_n_squared = np.sum(n_values**2) sum_nV = np.sum(n_values * V_values) # Calculating the slope (m) and intercept (b) n = len(n_values) m = (n * sum_nV - sum_n * sum_V) / (n * sum_n_squared - sum_n**2) b = (sum_V - m * sum_n) / n m, bThe linear regression equation that models the data is: $$V = 296.63n + 20250.93$$ Where: - $$V$$ is the value of the investment, - $$n$$ is the number of years after 1989. ### Based on the regression model: 1. **The value of the investment in 1989 (when $$n = 0$$):** Substitute $$n = 0$$ into the regression equation: $$V = 296.63(0) + 20250.93 = 20250.93$$ Thus, the value of the investment in 1989 was approximately **$20,250.93**. 2. **The rate of change per year (the slope $$m$$):** The rate of change is represented by the slope $$m$$, which is approximately **$296.63 per year**. ### Summary: - The value of the investment in 1989 was approximately **$20,250.93**. - The investment grows at a rate of **$296.63 per year**. Would you like further details or clarifications on any of the steps? Here are a few related questions for exploration: 1. How would the regression equation change if we had additional data points? 2. What is the meaning of the y-intercept in this context? 3. How can we assess the accuracy of this regression model? 4. What other statistical models might be better if the data isn't linear? 5. How does the choice of units (like dollars and years) affect the regression results? **Tip:** When using regression models, always check the residuals (differences between observed and predicted values) to verify the fit of the model!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Regression
Algebra
Statistics
Formulas
Linear regression equation: V = m * n + b
Slope formula: m = (n ΣV - Σn ΣV) / (n Σn^2 - (Σn)^2)
Intercept formula: b = (ΣV - m Σn) / n
Theorems
Regression analysis
Suitable Grade Level
Grades 11-12