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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.

The scatterplot above shows data on the average number of Americans who viewed the nightly news from

1980

2010

. The line of best fit is shown on the scatterplot. Based on the line of best fit, which of the following is the best estimate of the number of viewers in

1987

We must use the line of best fit to find the point corresponding to an

x

-value of

7

years since

1980

The coordinates of the point are about

(7, 43)

There were about

43

million viewers in

1987

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.

The scatterplot above shows data on the average number of Americans who viewed the nightly news from

1980

2010

. A line of best fit is shown on the scatterplot. Based on the line of best fit, which of the following is the best estimate of the number of viewers in

2015

We can extend the line of best fit and estimate the value of

y

when

x = 35

The coordinates of the point are about

(35, 15)

There were about

15

million viewers in

2015

An alternate approach is to use the slope and

y

-intercept to find the equation of the line of best fit and solve for

y

when

x = 35

. This approach may take longer and is only necessary if additional accuracy is required or if the desired point is far from the boundaries of graph provided.

The line of best fit passes through coordinates at about

(0, 50)

and

(5, 45)

The $y$ ‍-intercept is $50$ ‍.
The slope is $\frac{change in y}{change in x} = \frac{45 - 50}{5 - 0} = - 1$ ‍.

Therefore, the equation for the line of best fit is

y = - 1 x + 50

Substituting

x = 35

into this equation and solving for

y

gives us:

y = - 1 (35) + 50 = 15

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 $x = 0$ ‍.
The slope, $m$ ‍, is the change in $y$ ‍ when $x$ ‍ increases by $1$ ‍.

The equation for a line of best fit is:

y = m x + b

, where

(x, y)

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

(0, 100)

and

(13, 0)

, the slope of the line of best fit is about:

\begin{aligned} \frac{(100 - 0) percent}{(0 - 13) hours} & = \frac{100 percent}{- 13 hours} \\ \approx - 7.7 \frac{percent}{hour} \end{aligned}

This means that the battery life remaining decreases by about $7.7$ ‍ percent for every additional hour of time spent on phone.

The

y

-intercept is about

(0, 100)

This means when time spent on phone is $0$ ‍ hours, Paige has about $100$ ‍ percent of battery life remaining.

The

x

-intercept is about

(13, 0

This means when time spent on phone is about $13$ ‍ hours, Paige has about $0$ ‍ percent of battery life remaining.

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 = m (x) + b

, where

(x, y)

represents any point which satisfies this equation.

The $y$ ‍-intercept, $b$ ‍, is the $y$ ‍-value when $x = 0$ ‍.
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 $1$ ‍.

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This is very educational I dont have any questions
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How do you know which is the increase of money value and the increase of a rating for example?
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Course: Praxis Core Math > Unit 1

Interpreting linear models | Lesson

What is a linear model?

What skills are tested?

How can we estimate a value within the data shown?

How can we estimate a value beyond the data shown?

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

How do we interpret a linear model?

Your turn!

Things to remember

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