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6.4 Exponential Regression

Last updated 10 months ago
16 questions
Note from the author:
OBJECTIVES & STANDARDS
Math Objectives
  • Use technology to find the exponential function that best fits a data set
  • Analyze how well a regression model fits a data set using r2 values
  • Use regression models to summarize trends and make predictions
Common Core Math Standards
  • Link to all CCSS Math
  • CCSS.HSS.ID.B.6.A
  • CCSS.HSF.LE.A.2
  • CCSS.HSF.LE.B.5
Personal Finance Objectives
  • Analyze long-term trends in the stock market
  • Use exponential regression to summarize trends for different portfolios, funds, and asset types
National Standards for Personal Financial Education
Investing
  • 3b: Investigate the long-run average rates of returns on small-company stocks, large-company stocks, corporate bonds, and Treasury bonds
DISTRIBUTION & PLANNING
Distribute to students
  • Student Activity Packet
  • Application Problems
OBJECTIVES & STANDARDS
Math Objectives
  • Use technology to find the exponential function that best fits a data set
  • Analyze how well a regression model fits a data set using r2 values
  • Use regression models to summarize trends and make predictions
Common Core Math Standards
  • Link to all CCSS Math
  • CCSS.HSS.ID.B.6.A
  • CCSS.HSF.LE.A.2
  • CCSS.HSF.LE.B.5
Personal Finance Objectives
  • Analyze long-term trends in the stock market
  • Use exponential regression to summarize trends for different portfolios, funds, and asset types
National Standards for Personal Financial Education
Investing
  • 3b: Investigate the long-run average rates of returns on small-company stocks, large-company stocks, corporate bonds, and Treasury bonds
DISTRIBUTION & PLANNING
Distribute to students
  • Student Activity Packet
  • Application Problems
Intro/Warm-Up:
GRAPH: Fahad’s Investment
The data points below represent the value of Fahad’s investment account over the last 15 years. Fahad wants to summarize the trends in his account, so he finds the line of best fit.

1
Based on his regression, Fahad wrote the equation for the line of best fit:
y = 275x + 881.
  1. What is the slope of the line of best fit?_______
  2. What does the slope represent in this scenario?_______
1
Estimate Fahad’s investment value after 15 years…
a. Using the data points._______
b. Using the line of best fit._______
1

Do you think the line of best fit gives you a good prediction? Why or why not?

Learn It
Measuring Goodness of Fit
We can measure how well a model, like Fahad’s line of best fit, matches the data points given by using a statistical measure called r squared
Read the notes below and answer the questions that follow.

1

In your own words, what does r^2 (r squared) tell you?

1

Summarize: Looking at a graph, how might you know if a model had an r squared value close to 1?

Practice It
Part I:
Part II: Fahad’s Investments and r squared
Let’s return to Fahad’s investment from the start of the lesson. Fahad used a linear regression model to explain his investment’s value, but we could also use an exponential model. Compare the models and answer the questions.

1
Part 1:
Model A_______
Model B_______
Model C_______
1

What is the r^2 value of the linear model? What does that tell you?

1
Write the equation for the exponential model by substituting in the provided values for a and b. Round to the nearest hundredth._______
1
Predict Fahad’s investment value after 20 years…
a. Using the linear model.y=_______
b. Using the exponential model.y=_______
1

Which model is a better fit for the data set? Justify your response.

Desmos:
DESMOS: Exponential Regression in Investing
Link in google classroom
In earlier lessons, you learned how to find the line that best fits a data set using a linear regression model. We can do the same thing with exponential functions. Follow your teacher’s instructions to complete this Desmos activity which will be graded separetly in X2.
Application Problems
Part I: Robo Advisors
Robo advisors have been growing in popularity ever since they were first made available to the public approximately 15 years ago. The data points show how many billions of dollars were managed by robo-advisors each year since 2010 in the US. The curve is an exponential regression model for the data.

1
Based on the model, approximately how many assets did robo advisors have under management in 2015?_______ billion
1
Here are the results of the exponential regression. Write the equation for the exponential model by substituting the values for a and b into the equation. Round to the nearest hundredth.
_______
1
Based on the regression, what is the average annual growth rate of robo-advisor’s assets under management?_______ %
1
Based on the exponential regression model, predict the assets under management for robo-advisors in 2025._______
1

Do you think this model provides an accurate prediction for the future? Why or why not?

Level 2
12
Complete the table by writing the equation and r2 values based on the regression results. Round to the nearest hundredth.

Bond Fund Stock Fund
_______ _______

Equation:
_______ _______

1. Based on the regression model, what is the average annual rate of return of the Stock Fund?_______

2. Amar decides to invest $100 in the Stock Fund. He learns it has a total expense ratio of 0.03%. Write an equation to model the predicted value of his investment after x years._______

3. Based on the regression model, what is the predicted price of the Bond Fund after 15 years?_______
4. Why does the exponential function of best fit for the Bond Fund appear almost linear?_______
5. Which fund is more volatile?_______ Explain your reasoning._______
6. Which fund do you think would be a better long-term investment? _______ Why?_______