Assessing the Fit of a Linear Model

3.2.4 Assessing the Fit of a Linear Model

3.2.4 • Assessing the Fit of a Linear Model

Activity

The weight of ice cream sold at a small store in pounds ( $x$ ) and the average temperature outside in degrees Celsius ( $y$ ) are recorded in the table.

$x$	20	18	21	17	21.5	19.5	21	18
$y$	6	4.5	6.5	3.5	7.5	6.5	7	5

This table presents 8 corresponding data points for variables x and y.

For these data, create a scatter plot by hand on a coordinate grid and sketch a line that fits the data well.

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

The line you drew should go through the middle of the data in your scatter plot.

A graph shows a blue trend line with six green data points clustered closely around it, indicating a positive correlation. The grid is labeled with x and y axes.

Using Technology to Find the Equation of the Best Fit Line

Previously, you drew the best fit line by hand and found its equation without the use of technology. In this activity, and throughout the remainder of this lesson, you will use technology to find the equation of the best fit line. Click here for instructions on how to use technology to find the equation of the best fit line.

Now, use technology to create the scatter plot and compute the best fit line.

Compare your answer:

$y = 0.78 x - 9.44$

What is the value of the slope for the line of best fit? Round your answer to the nearest hundredth (two decimal places).

Compare your answer:

The slope is 0.78.

What does this slope value mean in this situation?

Compare your answer:

This means that about 1 additional pound of ice cream is sold with every 0.78-degree increase in the outside temperature.

What is the value of the $y$ -intercept for the line of best fit? Round your answer to the nearest hundredth (two decimal places).

Compare your answer:

The $y$ -intercept is $(0, - 9.44)$ .

What does this $y$ -intercept value mean in this situation?

Compare your answer:

This means that no ice cream is sold when the temperature is about −9.44 degrees Celsius.

Use the best-fit line to predict the $y$ -value when $x$ is 10.

Compare your answer:

About −1.64 degrees Celsius.

Is this a good estimate for the data? Be prepared to share your reasoning.

Compare your answer:

Your answer may vary, but here is a sample. This may be a good estimate for the data since the points seem close to the line, but something different might happen when the temperature gets below freezing, so it’s possible that when only 10 pounds of ice cream are sold, the relationship with the weather may be different.

Your teacher will give you a data table for one of the other scatter plots from the previous activity. Use technology and this table of data to create a scatter plot then determine the line of best fit.

Compare your answer:

Card A: $y = 1.1979 x + 1.3196$

Card B: $y = - 0.4337 x + 12.288$

Card C: $y = 0.9749 x + 4.6529$

Card D: $y = - 2.063 x + 16.144$

Card E: $y = 1.1979 x + 5.8196$

10.

Interpret the meaning of the value of the slope of this line of best fit.

Compare your answer:

Card A: $y = 1.1979 x + 1.3196$ . When $x$ is zero, $y$ is 1.3196. As $x$ increases by 1, $y$ increases, on average, by 1.1979.
Card B: $y = - 0.4337 x + 12.288$ . When $x$ is zero, $y$ is 12.288. As $x$ increases by 1, $y$ decreases, on average, by 0.4337.
Card C: $y = 0.9749 x + 4.6529$ . When $x$ is zero, $y$ is 4.6529. As $x$ increases by 1, $y$ increases, on average, by 0.9749.
Card D: $y = - 2.063 x + 16.144$ . When $x$ is zero, $y$ is 16.144. As $x$ increases by 1, $y$ decreases, on average, by 2.063.
Card E: $y = 1.1979 x + 5.8196$ . When $x$ is zero, $y$ is 5.8196. As $x$ increases by 1, $y$ increases, on average, by 1.1979.

Are you ready for more?

Extending Your Thinking

Priya uses several different ride services to get around her city. The table shows the distance, in miles, she traveled during her last 10 trips and the price of each trip, in dollars.

distance (miles)	price ($)
3.1	12.5
4.2	14.75
5	16
3.5	13.25
2.5	12
1	9
0.8	8.75
1.6	9.75
4.3	12
3.3	14

Priya creates a scatter plot of the data using the distance, $x$ , and the price, $y$ . She determines that a linear model is appropriate to use with the data. Use technology to find the equation of a line of best fit.

Compare your answer:

$y = 1.57 x + 7.6$

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

Compare your answer:

For example: The slope means that the fare increases by $1.57 for each additional mile driven. The $y$ -intercept means that the fee for getting in the car is approximately $7.60.

Use the line of best fit to estimate the cost of a 3.6-mile trip. Will this estimate be close to the actual value? Be prepared to show your reasoning.

Compare your answer:

For example: According to the line of best fit, the cost of a 3.6-mile trip is approximately $13.25, which is the same price as the 3.5-mile trip listed in the table.

On her next trip, Priya tries a new ride service and travels 3.6 miles, but she pays only $4.00 because she receives a discount. Include this trip in the table and calculate the equation of the line of best fit for the 11 trips. Did the slope of the equation of the line of best fit increase, decrease, or stay the same? Why? Be prepared to show your reasoning.

Compare your answer:

For example: The slope of the line decreased to $1.27 for each additional mile. It probably decreased because $4.00 is a far lower price per mile than any of her other trips because of the discount. This causes the slope, the increase in price for each additional mile, to decrease.

Priya uses the new ride service for her $12^{th}$ trip. She travels 4.1 miles and is charged $24.75. How do you think the slope of the equation of the line of best fit will change when this $12^{th}$ trip is added to the table?

Compare your answer:

For example: The slope of the line will probably increase because $24.75 is very expensive and will likely cause the cost for each additional mile to increase.

Video: Finding Lines of Best Fit

Watch the following video to learn more about how to find the line of best fit.

Access multimedia content

Self Check

Old Faithful is a geyser in Yellowstone National Park. The following table offers some rough estimates of the length of an eruption (in minutes) and the amount of water (in gallons) in that eruption.

Length (minutes)	1.5	2	3	4.5
Amount of Water (gallons)	3,700	4,100	6,450	8,400

Use technology to find an equation of the line of best fit. Round to the nearest hundredth.

$y=-1151.19x+1640.48$
$y = 1640.48x+1151.19$
$y = 800x +2500$
$y=1151.19x+1640.48$

Additional Resources

Technology for Best Fitting Lines

Scientists are interested in finding out how different species adapt to finding food sources. One group studied crocodilians to find out how their bite force was related to body mass and diet. The table below displays the information they collected on body mass (in pounds) and bite force (in pounds).

Species	Body mass (pounds)	Bite force (pounds)
Dwarf crocodile	35	450
Crocodile F	40	260
Alligator A	30	250
Caiman A	28	230
Caiman B	37	240
Caiman C	45	255
Crocodile A	110	550
Nile crocodile	275	650
Crocodile B	130	500
Crocodile C	135	600
Crocodile D	135	750
Caiman D	125	550
Indian gharial crocodile	225	400
Crocodile G	220	1,000
American crocodile	270	900
Crocodile E	285	750
Crocodile F	425	1,650
American alligator	300	1,150
Alligator B	325	1,200
Alligator C	365	1,450

Crocodilian Biting

The scatter plot below displays the data on body mass and bite force for the crocodilians in the study.

A scatter plot that shows the body mass in pounds on the x-axis and the bite force in pounds on the y-axis.

Using technology, the line of best fit for the data is $y = 3.02 x + 154.68$ .

The slope of the line of best fit is 3.02. This signifies that for every pound more the crocodile weighs, it has 3.02 more pounds of bite force.

The $y$ -intercept is 154.68, which doesn’t quite make sense for a crocodile that weighs 0 pounds, but it does give a minimum bite force to expect from a crocodile.

Try it

Try It: Technology for Best Fitting Lines

The following table gives the times of the gold, silver, and bronze medal winners for the men’s 100-meter race (in seconds) for the past 10 Olympic Games.

Use technology to write an equation of the line that represents the line of best fit for the years since 1976 and the mean times each year. Round to the nearest hundredth.

Year	2012	2008	2004	2000	1996	1992	1988	1984	1980	1976
Number of Years (since 1976)	36	32	28	24	20	16	12	8	4	0
Gold	9.63	9.69	9.85	9.87	9.84	9.96	9.92	9.99	10.25	10.06
Silver	9.75	9.89	9.86	9.99	9.89	10.02	9.97	10.19	10.25	10.07
Bronze	9.79	9.91	9.87	10.04	9.90	10.04	9.99	10.22	10.39	10.14
Mean Time	9.72	9.83	9.86	9.97	9.88	10.01	9.96	10.13	10.30	10.09

Compare your answer: Here is how to find the line of best fit.

$x$	36	32	28	24	20	16	12	8	4	0
$y$	9.72	9.83	9.86	9.97	9.88	10.01	9.96	10.13	10.30	10.09

This table presents a dataset of ten corresponding (x, y) numerical value pairs.

Create a scatter plot and line of best fit.

A scatter plot that shows the year on the x-axis and the mean time in seconds on the y-axis. The best fit line that is drawn decreases from left to right

The equation of the line of best fit, using technology, is $y = - 0.01 x + 10.2$ .