Math Problem Statement

Determine the linear regression equation that models the set of data above, and use this equation to answer the questions. The table shows the value, V, of an investment (in dollars) n years after 1988 for various values of n.

Solution

To solve this problem, we need to determine the linear regression equation that models the data and answers the questions based on that regression.

The table shows the following values:

nn (years after 1988)37121419
V(n)V(n) (investment value in dollars)101819548.1488678005.167334

The form of a linear regression equation is:

V(n)=mn+bV(n) = mn + b

Where:

  • mm is the slope (rate of decrease of the investment per year),
  • bb is the y-intercept (value of the investment in 1988, or when n=0n = 0).

Step 1: Find the slope mm

The slope is calculated using two data points (n1,V(n1))(n_1, V(n_1)) and (n2,V(n2))(n_2, V(n_2)) from the table. Let's choose the first and last data points:

m=V(n2)V(n1)n2n1=5927.0510181193m = \frac{V(n_2) - V(n_1)}{n_2 - n_1} = \frac{5927.05 - 10181}{19 - 3}

Step 2: Solve for the y-intercept bb

Once we have the slope, we can use one of the points to solve for bb by substituting it into the equation V(n)=mn+bV(n) = mn + b.

Let me calculate the slope and equation.The slope mm of the linear regression equation is approximately 265.87-265.87. This means the investment value decreases by about $265.87 per year.

Step 3: Find the y-intercept bb

Now, we will substitute one of the points (let's use n=3n = 3, V(3)=10181V(3) = 10181) into the equation V(n)=mn+bV(n) = mn + b to solve for bb:

10181=265.87(3)+b10181 = -265.87(3) + b

Let me calculate the y-intercept bb.The y-intercept bb is approximately 10978.6210978.62. This represents the estimated value of the investment in the year 1988 (when n=0n = 0).

The linear regression equation is:

V(n)=265.87n+10978.62V(n) = -265.87n + 10978.62

Step 4: Answer the questions

  1. The value of the investment in 1988 is approximately $10,978.62.
  2. The rate at which the value decreases is $265.87 per year.

Would you like to see more details, or do you have any other questions?

Here are 5 relative questions to expand on this information:

  1. What is the projected value of the investment in 10 years after 1988?
  2. How can we use this regression model to predict future values beyond the data points given?
  3. What happens to the investment value after 30 years, according to this model?
  4. How would the equation change if the investment's value increased instead of decreased?
  5. How accurate are linear regression models for predicting long-term trends in investments?

Tip: Always ensure to check if a linear regression model is the best fit by analyzing the trend in data.

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Math Problem Analysis

Mathematical Concepts

Linear Regression
Algebra
Slope-Intercept Form

Formulas

V(n) = mn + b
Slope formula: m = (V(n2) - V(n1)) / (n2 - n1)
Intercept formula: V(n) = mn + b

Theorems

Linear Regression Theorem

Suitable Grade Level

Grades 9-12