# Lesson 6: The Slope of a Fitted Line

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

## 6.1: Estimating Slope

Estimate the slope of the line.

## 6.2: Describing Linear Associations

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.

## 6.3: Interpreting Slopes

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

1. What is the slope of the line in the scatter plot for each situation?
2. What is the meaning of the slope in that situation?

$y=5,\!520.619x-1,\!091.393$

$y=\text-0.011x+40.604$

$y=0.59x-21.912$

## 6.4: Positive or Negative?

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?

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?

## 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.

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 $$w = 4.27h -37$$ where $h$ is the height of the dog and $w$ is the predicted weight of the dog.

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.

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.