Lesson 21The Slope of a Fitted Line

Learning Goal

Let’s look at how changing one variable changes another.

Learning Targets

  • I can use the slope of a line fit to data in a scatter plot to say how the variables are connected in real-world situations.

Lesson Terms

  • negative association
  • positive association

Warm Up: Estimating Slope

Problem 1

Estimate the slope of the line.

A graph of a line with points at (-4,-6), (5,2), (8,3)

Activity 1: Describing Linear Associations

Problem 1

For each scatter plot, decide if there is an association between the two variables, and describe the situation using one of these sentences:

  • For these data, as increases, tends to increase.

  • For these data, as increases, tends to decrease.

  • For these data, and do not appear to be related.

  1. A scatter plot of mileage on x vs price on y.
  2. A scatter plot of weight (carats) on x vs price (dollars) on y
  3. A scatter plot of high temperature (F) on x vs energy consumed (kwh) on y

Activity 2: Interpreting Slopes

Problem 1

For each of the situations, a linear model for some data is shown.

  • What is the slope of the line in the scatter plot for each situation?

  • What is the meaning of the slope in that situation?

  1. A scatter plot of weight (carats) on x vs price (dollars) on y with a line of best fit.

  2. A scatter plot of weight (kg) on x vs fuel efficiency (mpg) vs y with a line of best fit.

  3. A scatter plot of high temperature (F) on x vs energy consumed (kwh) on y with a line of best fit. Much of the data is off the line.

Are you ready for more?

Problem 1

The scatter plot shows the weight and fuel efficiency data used in an earlier lesson along with a linear model represented by the equation .

A scatter plot showing weight and fuel efficiency data with a negative correlation line of best fit drawn

  1. What is the value of the slope and what does it mean in this context?

  2. What does the other number in the equation represent on the graph? What does it mean in context?

  3. Use the equation to predict the fuel efficiency of a car that weighs 100 kilograms.

  4. Use the equation to predict the weight of a car that has a fuel efficiency of 22 mpg.

  5. Which of these two predictions probably fits reality better? Explain.

Activity 3: Positive or Negative?

Problem 1

For each of the scatter plots, decide whether it makes sense to fit a linear model to the data. If it does, would the graph of the model have a positive slope, a negative slope, or a slope of zero?

  1. A scatter plot with data in a negative correlation and mostly clustered in a line.
  2. A scatter plot with data in a mostly positive correlation.
  3. A scatter plot with a sale of 0 to 12 on x and -50 to 50 on y and data in a mostly negative correlation with a horizontal line of best fit at 0.
  4. A scatter plot with no correlation in data.
  5. A scatter plot with data mostly correlated but one outlier.

Problem 2

Which of these scatter plots show evidence of a positive association between the variables? Of a negative association? Which do not appear to show an association?

Lesson Summary

Here is a scatter plot that we have seen before. As noted earlier, we can see from the scatter plot that taller dogs tend to weigh more than shorter dogs. Another way to say it is that weight tends to increase as height increases. When we have a positive association between two variables, an increase in one means there tends to be an increase in the other.

A scatter plot of dog height (in) (horizontal from 6-30) vs dog weight (pounds) (vertical from 0-112). The data trends up and towards the right.

We can quantify this tendency by fitting a line to the data and finding its slope. For example, the equation of the fitted line is where is the height of the dog and is the predicted weight of the dog.

A scatter plot and line of best fit of dog height (in) (horizontal from 6-30) vs dog weight (pounds) (vertical from 0-112). The data trends up and towards the right.

The slope is 4.27, which tells us that for every 1-inch increase in dog height, the weight is predicted to increase by 4.27 pounds.

In our example of the fuel efficiency and weight of a car, the slope of the fitted line shown is -0.01.

A scatter plot of weight (kg) vs fuel efficiency (mpg). The line of best fit slopes down and to the right.

This tells us that for every 1-kilogram increase in the weight of the car, the fuel efficiency is predicted to decrease by 0.01 miles per gallon. When we have a negative association between two variables, an increase in one means there tends to be a decrease in the other.