# Lesson 10: Predicting Values

## Lesson 10: Predicting Values

### Objective:

Students will learn how to make predictions based on linear models.

### Materials:

1. What’s the Trend? handout (LMR_4.7_What’s the Trend) from lesson 8

2. Predicting Values handout (LMR_4.10_Predicting Values)

### Vocabulary:

observed value, predicted value

### Essential Concepts:

Essential Concepts:

The regression line can be used to make good predictions about values of y for any given value of x. This works for exactly the same reason the mean works well for one variable: the predictions will make your score on the mean squared residuals as small as possible.

### Lesson:

1. Entrance ticket: How do you find the equation of a line using two points?

Note to Teacher: Students may share their responses with a partner or you may choose to use this ticket as an assessment.

2. Reveal the equation of the line of best fit for the Arm Span vs. Height data and ask students to check their equations from the homework assignment:

$\widehat{height}=0.7328(armspan)+17.4957$

Note: Any time a hat is on top of a variable, this means we are making “predicted values” of that variable.

3. Whose equation came closest to the equation of the regression line? Ask the student whose equation came closest to share how he/she came up with the equation.

4. Inform students that the equation of the line is a rule that predicts the height based on a second variable, in this case, arm span.

5. Team discussion question:

Using the equation of the line of best fit provided, how can we predict the height of a student whose arm span is 67 inches?

6. Remind students that lines of best fit are also known as regression lines and they are models that can be used to make predictions. Today, they will explore more about this line.

7. Ask student teams to refer back to What’s the Trend? Handout (LMR_4.7). They should discuss the following questions and record their responses on the Predicting Values handout (LMR_4.10):

1. What do you notice about where the points are and where the line is? Some points are near the line, others are further away, and one point is exactly on the line. Data points are observed values and points on the line are predicted values.

2. Recall from Algebra that every line can be represented by an equation in the form y=mx+b. In this case, the equation of the regression line is y=3.2536x+154.3654. What do the x- and y-values represent in this equation? The x-values represent the number of explosions and the y-values represent the predicted profit.

3. According to the equation, what is the slope of this line? What does the slope mean in relation to the number of explosions? The slope is 3.2536. It is the rate of change between the number of explosions and the profit. It means that for every explosion increase the profit increases by 3.2536 dollars.

4. When the number of explosions (x-value) is zero, what is the profit (y-value)? How do you know? What does this mean? The profit is 154.3654 million dollars. Students may use the equation to show that they substituted zero for x, so the y-intercept is the profit. It means that if Michael Bay were to make a movie with NO explosions, this would be his projected profit.

5. If you wanted to know the profit for the point that lies the closest to the line, what would the equation be? Write the equation and solve it. Profit=3.2536(211)+154.3654. Profit=840.875 or 840,875,000 million dollars.

6. What was the actual profit for the point that lies closest to the line? The actual profit was 836,303,693 million dollars.

7. What if Michael Bay made a movie that had 325 explosions? What would his predicted profit be? Show how you arrived at the solution. By substituting 325 in the value of x in the equation, predicted profit will be \$1,211, 785, 400 or \$ 1, 211.7854, or by finding the point on the line or both.

8. Assign one question to each team for a share out. If two teams have the same question, one team will share its explanation first and the second team can agree, disagree, or add to the first team’s explanation.

Note: If students ask/wonder about the meaning of the R2, inform them that it is related to R, also known as the correlation coefficient. They will learn about R (not R2)in lesson 11.

### Class Scribes:

One team of students will give a brief talk to discuss what they think the 3 most important topics of the day were.

### Homework

Students will answer the following questions about the Scores Over Time plot (LMR_4.7):

• What do you notice about where the points are and where the line is?

• What do the y-and x-values represent in this equation?

• According to the equation, what is the slope of this line? What does the slope mean?

• When the x-value is zero, what is the y-value? How do you know? What does this mean?

• What would the predicted value of the score be if M. Night Shyamalan released a movie in 2015? How do you know?