Topic 2.6: Linear Regression Models

Learning Objectives:

Construct and read a simple linear regression model written in the form ŷ = a + bx.
Calculate a predicted response value using a linear regression model.
Interpret the roles of the slope, y-intercept, explanatory variable, and predicted response in context.
Explain that regression output gives predicted values, not exact observed values.
Determine when a prediction is extrapolation and judge whether that prediction is reliable.

Essential Knowledge:

A simple linear regression model is an equation that uses an explanatory variable, x, to predict a response variable, y.
The predicted response is written as ŷ.
In the model ŷ = a + bx, a is the y-intercept and b is the slope.
Technology is commonly used to generate and use a regression model.
Extrapolation means predicting a response using an x-value outside the interval used to build the regression line.
Predictions made by extrapolation are less reliable because the trend may not continue beyond the observed data range.

Key Vocabulary

Linear Regression Model	An equation that uses an explanatory variable to predict a response variable.
Predicted Response	The model's estimated y-value, written as ŷ.
Slope	The amount the predicted response changes for each one-unit increase in x.
Y-Intercept	The predicted response when x = 0.
Explanatory Variable	The x-variable used to help predict the response.
Response Variable	The y-variable being predicted.
Regression Line	The line given by the linear regression model.
Prediction	An estimated value from the model rather than an exact observed value.
Observed Value	An actual value recorded in the data set.
Extrapolation	Using the model to predict beyond the interval of observed x-values.
Reliability	How trustworthy a prediction is likely to be.
Interval of x-values	The range of explanatory-variable values used to create the model.

Video 1: Building and Using a Regression Model

[0:00 – 5:53]

1. [0:00] Topic 2.6 introduces the components of the linear regression and shows how to make using it.

2. [0:13] The lesson goals are to construct a linear regression model, make , and gauge the of those predictions.

Healthy Food Access Context [0:25 – 1:48] Linda Salcedo studied 37 grocery stores in the San Antonio metropolitan area. She recorded the average income in each store's zip code and the number of organic vegetable items offered so she could investigate whether neighborhood income helps predict access to healthy foods.

3. [1:22] Linda's data set included grocery stores.

4. [1:48] In this study, average zip code was the explanatory variable and the number of organic was the response variable.

5. [2:22] The statistics version of a line is the linear regression equation .

Linear Regression Equation [2:22 – 3:15]

ŷ = a + bx

ŷ is the predicted response value.
b is the slope of the regression line.
a is the y-intercept of the regression line.
Technology is usually used to find the model.

6. [2:22] In the regression equation, ŷ is the y-value, b is the , and a is the .

7. [3:15] Students usually use to find a regression model such as ŷ = + x.

8. [3:42 – 4:12] To predict for a zip code with average income of $90,000, we substitute 90,000 for and get a predicted response of .

9. [4:38] The answer 75.3 can be a decimal because it is a prediction, not an store value.

10. [5:36] The linear regression model is composed of a slope and a , and its predictions do not represent data values.

11. Reflection: How do slope, y-intercept, and x-value work together in ŷ = a + bx, and why is the result only a prediction?

Video 2: Extrapolation and Reliability

[0:00 – 10:12]

2048 Prediction Example [0:34 – 2:29] The video revisits a study that used a linear model to predict that 100% of Americans would be overweight by 2048. This example is used to show why predictions made beyond the observed x-values can be risky.

12. [0:34] The study predicted that by 2048, of Americans would be overweight if the trend continued.

13. [1:14] Making predictions outside the interval of observed is called .

14. [1:54] Extrapolation can be dangerous because the trend may not .

Extrapolation Warning [1:14 – 2:29]

Extrapolation means predicting outside the interval of observed x-values.
Predictions made by extrapolation are less reliable.
The farther we predict beyond the data range, the more cautious we should be.
This warning applies to time variables and non-time variables.

15. [2:29] The warning about extrapolation applies to non-time explanatory variables as .

16. [7:42] In the swine population example, because x was measured in thousands, a population of 200 had to be written as .

17. [8:15] Using the regression equation gave a predicted ammonia concentration of .

18. [8:15] Because x = 0.2 was outside the observed interval of x-values, the prediction was .

19. [8:53] The model response says that prediction was not reliable due to .

20. Reflection: Why is extrapolation dangerous, and how do you decide whether a regression prediction is outside the observed data range?

Exit Ticket

A teacher uses the linear regression model ŷ = 61.4 + 1.8x to predict a student's quiz score, where x is the number of hours of after-school tutoring in one week. The model was built using data from students who tutored between 2 and 10 hours.

(a) Identify the explanatory variable, the response variable, the slope, and the y-intercept in context.

(b) Calculate the predicted quiz score for a student who tutors 8 hours.

(c) Explain why the predicted score does not have to match a student's exact quiz score.

(d) Would using the model for a student who tutors 15 hours be reliable? Explain.

Topic 2.6: Linear Regression Models

Key Vocabulary

Video 1: Building and Using a Regression Model

Video 2: Extrapolation and Reliability

Exit Ticket

Class Responses (Esc to close)