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# Interpreting linear models | Lesson

## What is a linear model?

If we graph data and notice a trend that is approximately linear, we can model the data with a line of best fit. A line of best fit can be estimated by drawing a line so that the number of points above and below the line is about equal.
While a line of best fit is not an exact representation of the actual data, it is a useful model that helps us interpret the data and make estimates.

### What skills are tested?

• Using a line of best fit to estimate values within or beyond the data shown
• Identifying the equation of a line of best fit
• Interpreting the slope of a line of best fit in a real world context

## How can we estimate a value within the data shown?

We can use a line of best fit to estimate a value within the data shown. Estimating a value means finding a y-value when given a specific x-value or finding an x-value when given a specific y-value on the line of best fit
To estimate a value within the data shown, use the graph scales to locate the desired point on the line of best fit, and then estimate the other coordinate.

## How can we estimate a value beyond the data shown?

We can use a line of best fit to estimate a value beyond the data shown. Estimating a value means finding a y-value when given a specific x-value or finding an x-value when given a specific y-value on the line of best fit.
To estimate a value beyond the data shown, extend the graph scale and line of best fit to include the desired point and then estimate the value of the other coordinate.

## How do we determine the equation of a line of best fit?

A line of best fit usually shows two key features.
• The y-intercept, b, is the y-value when x, equals, 0.
• The slope, m, is the change in y when x increases by 1.
The equation for a line of best fit is: y, equals, m, x, plus, b, where left parenthesis, x, comma, y, right parenthesis represents any point that satisfies this equation.

## How do we interpret a linear model?

In context, the meaning of the slope and intercepts of the line of best fit must be explained with the appropriate units.
For example, suppose Paige collected data on how much time she spent on her phone and the percent battery life remaining. The scatterplot below shows the data and the line of best fit.
Using the points left parenthesis, 0, comma, 100, right parenthesis and left parenthesis, 13, comma, 0, right parenthesis, the slope of the line of best fit is about:
\begin{aligned} \dfrac{(100-0)\text{ percent}}{(0-13)\text{ hours}} &=\dfrac{100\text{ percent}}{-13 \text{ hours}} \\\\ &\approx -7.7\, \dfrac{\text{percent}}{\text{hour}} \end{aligned}
• This means that the battery life remaining decreases by about 7, point, 7 percent for every additional hour of time spent on phone.
The y-intercept is about left parenthesis, 0, comma, 100, right parenthesis.
• This means when time spent on phone is 0 hours, Paige has about 100 percent of battery life remaining.
The x-intercept is about left parenthesis, 13, comma, 0).
• This means when time spent on phone is about 13 hours, Paige has about 0 percent of battery life remaining.

TRY: ESTIMATING WITHIN THE DATA SHOWN
A teacher wants to determine if there is a relationship between students' scores on their first exam and their second exam. The scatterplot above shows her data and the line of best fit. Based on the line of best fit, for a student whose first exam score is 80, what is a reasonable estimate of their score on the second exam?

TRY: ESTIMATING BEYOND THE DATA SHOWN
The scatterplot above shows data from a random sample of people who reported the age and mileage of their cars, along with the line of best fit. Based on the line of best fit, what is a reasonable estimate of the mileage, in thousands of miles, of a car that is 13 years old?

TRY: IDENTIFYING THE EQUATION OF THE LINE OF BEST FIT
Which of the following could represent the equation of the line of best fit for the scatterplot shown above?

TRY: INTERPRETING THE SLOPE
A panel is rating different kinds of potato chips. The scatterplot above shows the relationship between their average rating and the price of the chips, along with the line of best fit. The slope of the line of best fit is 5. What is the best interpretation of this slope?

## Things to remember

A line of best fit can be estimated by drawing a line so that the number of points above and below the line is about equal.
We can use a line of best fit to estimate values within or beyond the data shown.
• To estimate a value within the data shown, use the graph scales to locate the desired point on the line of best fit, and then estimate the other coordinate.
• To estimate a value beyond the data shown, extend the graph scale and line of best fit to include the desired point, and then estimate the value of the other coordinate.
The equation for a line of best fit is: y, equals, m, left parenthesis, x, right parenthesis, plus, b, where left parenthesis, x, comma, y, right parenthesis represents any point which satisfies this equation.
• The y-intercept, b, is the y-value when x, equals, 0.
• The slope, m, is the change in y when x increases by 1.
In context, the meaning of the slope of a line of best fit must be explained with the appropriate units.
• The slope specifies the change in y when x increases by 1.

## Want to join the conversation?

• This is very educational I dont have any questions
(1 vote)
• How do you know which is the increase of money value and the increase of a rating for example?
(1 vote)