Interpreting the Slope and Vertical Intercept of a Scatter Plot - Algebra 1

3.1.4 Interpreting the Slope and Vertical Intercept of a Scatter Plot

3.1.4 • Interpreting the Slope and Vertical Intercept of a Scatter Plot

Activity

Here are several scatter plots.

Use scatter plot A to answer questions 1 – 2.

A scatter plot with a trend line that indicates a negative correlation. The x-axis represents age in years and extends from 0 to 24 with a scale of 2. The y-axis represents reaction time in milli-seconds and extends from 0 to 350 with a scale of 25.

Scatter plot A: $y = - 9.25 x + 400$

1.

Using the horizontal axis for $x$ and the vertical axis for $y$ , interpret the slope of the linear model in the situation shown.

Compare your answer: Your answer may vary, but here is a sample.

For every increase in age of one year, the linear model estimates that the reaction time decreases by about 9.25 milliseconds.

2.

If the linear relationship continues to hold for the situation, interpret the $y$ -intercept of the linear model.

Compare your answer: Your answer may vary, but here is a sample.

The reaction time estimated by the linear model for a newborn is 400 milliseconds.

Use scatter plot B to answer questions 3 and 4.

Scatter plot B: $y = 0.44 x + 0.04$

A scatter plot with a trend line that indicates a positive correlation. The x-axis represents number of bananas and extends from 0 to 7 with a scale of 1. The y-axis represents price in dollars and extends from 0 to 3.50 with a scale of 0.5.

3.

Using the horizontal axis for $x$ and the vertical axis for $y$ , interpret the slope of the linear model in the situation shown.

Compare your answer: Your answer may vary, but here is a sample.

For every additional banana, the linear model estimates that the price increases by about $0.44.

4.

If the linear relationship continues to hold for the situation, interpret the $y$ -intercept of the linear model.

Compare your answer: Your answer may vary, but here is a sample.

The price estimated by the linear model for purchasing zero bananas is $0.04.

Use scatter plot C to answer questions 5 – 6.

Scatter plot C: $y = 4 x + 87$

A scatter plot with a trend line that indicates a positive correlation. The x-axis represents room size in square feet and extends from 0 to 500 with a scale of 50. The y-axis represents the cost to install flooring in dollars and extends from 0 to 2,200 with a scale of 100.

5.

Using the horizontal axis for $x$ and the vertical axis for $y$ , interpret the slope of the linear model in the situation shown.

Compare your answer: Your answer may vary, but here is a sample.

For each additional square foot, the linear model estimates that the cost to install flooring increases by about $4.

6.

If the linear relationship continues to hold for the situation, interpret the $y$ -intercept of the linear model.

Compare your answer: Your answer may vary, but here is a sample.

The fixed cost to install flooring is $87.

Use scatter plot D to answer questions 7 and 8.

Scatter plot D: $y = - 2.4 x + 25.0$

A scatter plot with a trend line that indicates a negative correlation. The x-axis represents the temperature in degrees Celsius and extends from 0 to 5.5 with a scale of 0.5. The y-axis represents the volume in cubic centimeters and extends from 0 to 24 with a scale of 2.

7.

Using the horizontal axis for $x$ and the vertical axis for $y$ , interpret the slope of the linear model in the situation shown.

Compare your answer: Your answer may vary, but here is a sample.

For every increase in temperature of 1 degree Celsius, the linear model estimates that the volume decreases by about 2.4 cubic centimeters.

8.

If the linear relationship continues to hold for the situation, interpret the $y$ -intercept of the linear model.

Compare your answer: Your answer may vary, but here is a sample.

25 cubic centimeters is the volume estimated by the linear model for a temperature of zero degrees Celsius.

Are you ready for more?

Extending Your Thinking

Clare, Diego, and Elena collect data on the mass and fuel economy of cars at different dealerships.

Clare finds the line of best fit for data she collected for 12 used cars at a used car dealership. The line of best fit is $y = \frac{- 9}{1, 000} x + 34.3$ where $x$ is the car’s mass, in kilograms, and $y$ is the fuel economy, in miles per gallon.

Diego made a scatter plot for the data he collected for 10 new cars at a different dealership.

A scatter plot with a negative correlation. The x-axis represents mass measured in kilograms and extends from 1,200 to 2,400 with a scale of 200. The y-axis represents the fuel economy in miles per gallon and extends from 0 to 30 with a scale of 5.

Elena made a table for data she collected on 11 hybrid cars at another dealership.

mass (kilograms)	fuel economy (miles per gallon)
1,100	38
1,200	39
1,250	35
1,300	36
1,400	31
1,600	27
1,650	28

1.

Interpret the slope and $y$ -intercept of Clare’s line of best fit in this situation.

Compare your answer:

The slope means that the fuel economy decreases 9 miles per gallon for every 1,000 kg increase in mass. The $y$ -intercept represents the fuel economy of a car, in miles per gallon, that has a mass of 0 kg.

2.

Diego looks at the data for new cars and used cars. He claims that the fuel economy of new cars decreases as the mass increases. He also claims that the fuel economy of used cars increases as the mass increases. Do you agree with Diego’s claims? Be prepared to show your reasoning.

Compare your answer:

Diego’s claim about new cars is correct. You can look at the graph and see that the fuel economy tends to decrease as the mass increases because the scatter plot appears to have a negative slope. Diego’s claim about used cars is not correct because the line of best fit has a negative slope, which means that the fuel economy decreases as the mass increases.

3.

Elena looks at the data for hybrid cars and correctly claims that the fuel economy decreases as the mass increases. How could Elena compare the decrease in fuel economy as mass increases for hybrid cars to the decrease in fuel economy as mass increases for new cars? Be prepared to show your reasoning.

Compare your answer:

Elena needs to compare the slope of the data for hybrids to the slope of the data for new cars. She could do this by looking at the scatter plots and graphing an approximated line of best for each scatter plot. She could then see which line of best fit looks like it is decreasing faster or find and compare the slopes of each approximated line of best fit.

Self Check

The scatter plot below shows the results of a survey of eighth-grade students who were asked to report the number of hours per week they spend playing video games and the typical number of hours they sleep each night.

A SCATTER PLOT THAT SHOWS VIDEO GAME TIME IN HOURS PER WEEK ON THE X-AXIS AND SLEEP TIME IN HOURS PER NIGHT ON THE Y-AXIS. THE LINE DRAWN DECREASES FROM LEFT TO RIGHT.

Which of the following conclusions can you make looking at the data?

There is not a trend between time spent on video games and sleep time.
As video game time increases, sleep time decreases.
As video game time decreases, sleep time decreases.
As video game time increases, sleep time increases.

Additional Resources

Using Linear Models

The Monopoly board game is popular in many countries. The scatter plot below shows the distance from “Go” to a property (in number of spaces moving from “Go” in a clockwise direction) and the price of the properties on the Monopoly board. The equation of the line is $P = 8 x + 40$ , where $P$ represents the price (in Monopoly dollars) and $x$ represents the distance (in number of spaces).

A scatter plot that shows the "distance from go" in number of spaces on the x-axis and the price of a property in monopoly dollars on the y-axis. The line drawn increases from left to right.

The model shows that as the distance from “Go” increases, the price of the property increases.

The slope of the equation of the linear model is 8. This means that for every space from “Go,” the price of the property increases $8.

Notice this gives an estimate. The most expensive property is 39 spaces from “Go” and costs $400. The model would give the price to be $y = 8 (39) + 40$ or $352.

The model gives a difference of $48 between the estimated price using the model and the actual price.

Try it

Using Linear Models

A scatter plot that shows the mean temperature, in degrees Fahrenheit, in July on the x-axis and the mean rainfall per year, in inches, on the y-axis. The line drawn increases from left to right. The x-axis representing the number of people extends from 65 to 80 with a scale of 0.5. The y-axis representing the noise level extends from 30 to 42 with a scale of 0.5.

Looking at the linear model above, what claim can be made?

Compare your answer:

Here is one claim that can be made looking at the scatter plot and linear model:

As the mean July temperature of a city increases, so does the mean rainfall per year.