Unit 8 Section 4 : Lines of Best Fit

When there reasonable correlation between two variables on a scatter plot, it is possible to draw a line of best fit. This line represents the underlying relationship between the two quantities. When drawing a line of best fit the aim is to keep the distances of all the points from the line to a minimum. Sometimes it is helpful to try to keep the number of points above the line the same as the number of points below the line.

Lines of best fit can be used to make predictions. The accuracy and reliability of those predictions will depend on the strength of the correlation between the two variables.

Example 1

Draw a line of best fit for the points in the following scatter graph:

(a)

Use the data to draw a scatter graph.

Note that there are 3 points above the line and 3 below. The total distances to the points above the line is similar to the total distance to the points below the line.
(b)

Use the line to predict the value of y when x = 12.

Using the dotted line, we have y = 6.4 when x = 12.

Example 2

The following data was collected from an experiment. In the experiment, objects of different masses were placed on a horizontal surface and the force needed to make them start to move was recorded.

Mass(kg) 0.5 1.0 1.5 2.0 3.0 5.0
Force(N) 2.1 3.8 6.1 7.9 13.2 19.1

Use a scatter graph to estimate the force needed for a 2.5 kg mass.

The graph also shows that the estimated force for a 2.5 kg mass is 10 N.

Exercises

Question 1

x 1 2 3 4 5 6
y 7 10 12 15 19 21
(a)

Use the data shown to draw a scatter plot and draw a line of best fit for the data.

You need to upgrade your Flash Player
Go to http://www.adobe.com/go/getflashplayer.
(b)

Estimate the value of y when x = 0.

y =
Question 2

The following table lists values of x and y.

Pupil A B C D E F G H I J
Maths Score 45 83 65 62 71 52 69 72 58 64
Science Score 39 80 59 60 65 54 65 67 56 64
(a)

Draw a scatter graph for this data and then draw a line of best fit.

You need to upgrade your Flash Player
Go to http://www.adobe.com/go/getflashplayer.
(b)

Estimate the score on the Science test for pupils who scored:

(i) 73 About
(ii)40 About

on the Maths test.

Question 3

The following data was collected by a lorry driver who was interested in how much fuel he used on different journeys.

Length of Journey(miles) 100 250 150 180 220 300
Fuel Used(litres) 24 59 44 50 59 97
(a)

Draw a scatter graph and a line of best fit for this data.

You need to upgrade your Flash Player
Go to http://www.adobe.com/go/getflashplayer.
(b)

Estimate how much fuel would be needed for a 200 mile journey.

About litres
Question 4

A pupil carried out an experiment where he recorded the length of a spring when various masses were hung from it.

Mass (grams) 5050100150200300
Length (cm) 6.06.66.98.09.111.1

Use a scatter graph and a line of best fit to estimate the length of the spring when:

You need to upgrade your Flash Player
Go to http://www.adobe.com/go/getflashplayer.
(a)

no mass is hung from it,

About cm
(b)

a mass of 250 grams is hung from it.

About cm
Question 5

Rafiq collected the following data on the height and shoe size of some pupils in his class:

Shoe Size 648591045.5
Height (cm) 143150172146165177141156
(a)

Draw a scatter plot and a line of best fit for the data.

You need to upgrade your Flash Player
Go to http://www.adobe.com/go/getflashplayer.
(b)

Estimate the height of a person with a shoe size of 7.5.

About cm
(c)

Ian has a height of 170 cm. Estimate his shoe size.

About
Question 6

A garage owner keeps a record of the age and price of the small family cars that the garage sells. Some of these records are given in the following table:

Age(years) 65731237910
Price(£) 5700680053007700850079007800570037003600
(a)

Draw a scatter graph and a line of best fit for this data.

You need to upgrade your Flash Player
Go to http://www.adobe.com/go/getflashplayer.
(b)

Estimate the price of a 4-year-old car and a 12-year-old car.

4-year-old: about £
12-year-old: about £
Question 7

An electric heater was turned on in a cold room. The temperature was recorded every 2 minutes.

Time (minutes) 02468101214161820
Temperature (°C) 8.09.310.411.512.713.915.016.017.018.219.4
(a)

Estimate the temperature after 15 minutes.

About °C
(b)

Estimate when the temperature will reach 22 °C.

About minutes
Perfect positive correlation
Question 8

A biology student measured the height of a small plant at weekly intervals.
The results obtained are listed in the following table:

Time (weeks) 01234567
Height (cm) 1.22.53.64.55.36.47.28.3
(a)

Estimate the height of the plant after 3.5 weeks.

About cm
(b)

Estimate when the height of the plant will be 10 cm.

About weeks
Question 9

The scatter diagram shows the heights and masses of some horses. The scatter diagram also shows a line of best fit.

(a)

What does the scatter diagram show about the relationship between the height and mass of the horses?

The taller the horse, the its mass.
(b)

The height of a horse is 163 cm. Use the line of best fit to estimate the mass of the horse.

Approximately kg
(c)

A different horse has a mass of 625 kg. Use the line of best fit to estimate the height of the horse.

Approximately cm
(d)

A teacher asks his class to investigate this statement: "The length of the back leg of a horse is always less than the length of the front leg of a horse." What might a scatter graph look like if this statement is correct?
Show your answer on a copy of the axes below.

You need to upgrade your Flash Player
Go to http://www.adobe.com/go/getflashplayer.
Question 10

Nine students were discussing their holiday jobs working on a local farm. They decided to find out if there were any relationships between the time they spent working, sleeping, watching television and the distance they had to travel to work. The students plotted three scatter graphs.

(a)

What does Graph 1 show about the relationship between the weekly hours spent watching television and the weekly hours worked?

Fewer hours worked – time spent watching television.
(b)

What does Graph 2 show about the relationship between the weekly hours slept and the weekly hours worked?

in hours slept for increased working time.
(c)

What does Graph 3 show about the relationship between the weekly travelling distance and the weekly hours worked?

relationship between weekly travelling distance and hours worked.
(d)

Another student works 30 hours per week. Use Graph 1 to estimate the weekly hours spent watching television by this student.

About hours