What to Know About 6.E: Calculating Predicted Values of the Response Variable

Learning Goals

At the end of this page, you should feel comfortable performing these skills:

  • Use a scatterplot to describe bivariate relationships.
  • Approximate predicted values from a scatterplot.
  • Calculate predictions using the line of best fit.
  • Assess reliability of a prediction calculated using the line of best fit.

For the upcoming Forming Connections activity, you will need to use technology to make a scatterplot, calculate the line of best fit, and use the line to calculate predicted values of the response variable. Let’s walk through a scenario to practice those skills before you dive into that activity. The questions on this page will provide a refresher for the ideas you’ve learned previously in this module, so it should feel fairly comfortable. As such, there is no video demonstration for these questions. Use this page to assess your understanding before you finish this module.

Linear Analysis

What is the relationship between the size and price of a house in Florida? To answer this question, you will analyze data for 100 houses in Gainesville, Florida that sold in 2003. The data are from the “House Prices in FL” dataset available in the DCMP Linear Regression tool at https://dcmathpathways.shinyapps.io/LinearRegression/.

The data contain the sizes of the homes (in square feet) and prices of the homes (in thousands of U.S. dollars).

question 1

The goal of this analysis is to predict the price of a house based on its size. Identify the explanatory and response variables.

Visualizing the Relationship

question 2

Use the tool to make a scatterplot to visualize the relationship between the two variables. Describe the relationship between the two variables. Select all that apply.

  1. a) Positive
  2. b) Negative
  3. c) Quadratic
  4. d) Linear
  5. e) No relationship
  6. f) Weak
  7. g) Strong

Use the scatterplot from Question 2 to answer Questions 3 and 4. Don’t calculate and fit a regression line yet. You’ll do that in Question 5.

Approximating From a Scatterplot

question 3

Suppose there is a house for sale that has a size of 1,500 square feet. Based on the scatterplot, which of the following is the most likely value for its price?

  1. a) $100,000
  2. b) $110,000
  3. c) $125,000
  4. d) $150,000

 

question 4

Suppose you fit a line of best fit for these data. The line would predict the price of a house that has a size of 1,000 square feet to be closest to what value?

  1. a) $90,000
  2. b) $100,000
  3. c) $110,000
  4. d) $120,000

 The Regression Line Equation

question 5

Use technology to calculate the line of best fit and write the equation for the line.

 

Part A: Write the equation using appropriate notation and customize the names of the variables.

 

Part B: Interpret the slope, in context. Select the best answer.

a) For every one square foot increase in the size, there will be an increase of $77.01 in the price of a house in Florida.

b) For every one square foot increase in the size, we predict an average increase of $77.01 in the price of a house in Florida.

c) For every $1 increase in the price, there will be an increase of 77.01 square feet in the size of a house in Florida.

d) For every $1 increase in the price, we predict an average increase of 77.01 square feet in the size of a house in Florida.

 

Part C: Does the intercept have a meaningful interpretation? Select the best answer.

a) Yes. At a size of zero square feet, the predicted price of a house in Florida is $9,161.

b) Yes. At a price of $0, the predicted size of a house in Florida is 9,161 square feet.

c) No. It does not make sense to have a house that is zero square feet.

d) No. It does not make sense to have a house that costs $0.

 

Part D: Use the line to calculate the predicted price for a house that is 1,852 square feet. Round your answer to 2 decimal places.

 

Part E: Use the line to calculate the predicted price for a house that is 2,860 square feet. Round your answer to 2 decimal places.

 

Part F: Do you think the model prediction is more reliable for houses that are 1,852 square feet or for those that are 2,860 square feet? Explain.

Summary

In this What to Know page, you practiced the basic skills necessary to perform a linear regression analysis. Let’s summarize these skills.

  • In Question 1, you identified the explanatory and response variables for a given scenario.
  • In Question 2, you used technology to make a scatterplot.
  • In Question 2, you also used a scatterplot to describe bivariate relationships.
  • In Questions 3 and 4, you approximated predicted values from a scatterplot.
  • In Question 5, Part A, you used technology to calculate a line of best fit.
  • In Question 5, Parts B and C, you interpreted the slope and intercept of the line of best fit.
  • In Question 5, Parts D through F, you calculated predictions using the line of best fit and assessed reliability of the predictions.

This page provided a summary and practice of the ideas you’ve learned in this section (and in [Section 5A] previously). Feel free to return to it any time to practice. Hopefully, it felt comfortable and familiar. Let’s move on to Forming Connections to put it all together.