6. Lines of Fit
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Chapter 4
6.
Lines of Fit
In the lesson, the concept of residuals is introduced as the vertical distance between a data point and a line of fit on a scatter plot. The line of fit is essentially a line that best represents a set of data points. To evaluate how well this line fits the data, the sum of squared residuals is calculated. The lower this sum, the better the line models the data. Scatter plots are used to visualize these concepts, helping you understand the relationship between variables. For example, if you are researching used car prices based on mileage, you can plot this data and draw a line of fit to make better predictions. The sum of squared residuals will tell you how reliable your predictions are likely to be.
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| | 8 Theory slides |
| | 8 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Lines of Fit
Slide of 8
There are several ways to analyze how well a line of fit models a data set. In this lesson, the differences between observed data points and the points predicted by the line of fit will be calculated to determine if the regression line models the data well.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Explore
Scatter Plot of Bivariate Data and Line of Fit
Consider the scatter plot of bivariate data. Move the the line in such a way that it models the data. 
For the different data sets, draw the line of fit. How can it be determined if this line models the data well?
Discussion
Residual
One way to assess how well a line of fit describes the data is to analyze residuals.
Concept
Residual
A residual is the vertical distance between a data point and the line of fit. When a line of fit has been drawn on a scatter plot, not all of the data points lie exactly on the line — some of them are above the line and some below. Therefore, each data point has one residual, which can be positive, negative, or zero.
A residual can also be defined as the observed y-value of a data point minus its predicted y-value, found using the line of fit.
Residual=Observed y-value−Predicted y-value
- If the points in the plot are randomly placed about the x-axis, then a linear model describes the data set well.
- If some kind of pattern appears in the scatter plot, a non-linear model is more appropriate for the data.
Pop Quiz
Practice Finding Sum of Squared Residuals
The applet generates bivariate data and the equation of a line of fit for the data. Calculate the sum of the squares of the residuals for the given equation.
Example
Determining Which Line of Fit Is Better
The table below shows the finishing times, in seconds, for the Olympic gold medalist in the men's 100-meter dash for the last six Olympic games. Olympic Game 1 represents the 2000 Olympic games and Olympic Game 6 represents the 2020 Olympic games.
| Olympic Game | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Finishing Time (sec) | 9.87 | 9.85 | 9.69 | 9.63 | 9.81 | 9.80 |
Equation I:Equation II: y=-0.1x+10 y=-0.05x+10
In these equations, x represents the Olympic Game number as described above, and y represents the finishing time in seconds. For each equation, calculate the sum of squared residuals. Which of the equations is a better fit?
b Make a scatter plot for the residuals.
Answer
a Sum of Squared Residuals for Equation I: 0.2605
Sum of Squared Residuals for Equation II: 0.077
b Graph:
Hint
a Make a table of the residual values for each equation and then find the sum of the squares of the residuals.
b Plot the points (x,residual) on a coordinate plane.
Solution
a For the given data set, there are two possible lines of fit.
Equation I:Equation II: y=-0.1x+10 y=-0.05x+10
In order to determine which line of fit is better, the residuals for both lines will be calculated first. For a data point, its residual is the difference between the y-value of the data point and the y-value predicted by the line of fit. | x | y (Actual) | y Predicted by y=-0.1x+10 | Residual for y=-0.1x+10 |
|---|---|---|---|
| 1 | 9.87 | y=-0.1(1)+10=9.9 | 9.87−9.9=-0.03 |
| 2 | 9.85 | y=-0.1(2)+10=9.8 | 9.85−9.8=0.05 |
| 3 | 9.69 | y=-0.1(3)+10=9.7 | 9.69−9.7=-0.01 |
| 4 | 9.63 | y=-0.1(4)+10=9.6 | 9.63−9.6=0.03 |
| 5 | 9.81 | y=-0.1(5)+10=9.5 | 9.81−9.5=0.31 |
| 6 | 9.80 | y=-0.1(6)+10=9.4 | 9.80−9.4=0.40 |
(-0.03)2+0.052+(-0.01)2+0.032+0.312+0.402
CalcPow
Calculate power
0.0009+0.0025+0.0001+0.0009+0.0961+0.16
AddTerms
Add terms
0.2605
| x | y (Actual) | y Predicted by y=-0.05x+10 | Residual for y=-0.05x+10 |
|---|---|---|---|
| 1 | 9.87 | y=-0.05(1)+10=9.95 | 9.87−9.95=-0.08 |
| 2 | 9.85 | y=-0.05(2)+10=9.9 | 9.85−9.9=-0.05 |
| 3 | 9.69 | y=-0.05(3)+10=9.85 | 9.69−9.85=-0.16 |
| 4 | 9.63 | y=-0.05(1)+10=9.8 | 9.63−9.8=-0.17 |
| 5 | 9.81 | y=-0.05(5)+10=9.75 | 9.81−9.75=0.06 |
| 6 | 9.80 | y=-0.05(6)+10=9.70 | 9.80−9.70=0.10 |
(-0.08)2+(-0.05)2+(-0.16)2+(-0.17)2+0.062+0.102
CalcPow
Calculate power
0.0064+0.0025+0.0256+0.0289+0.0036+0.01
AddTerms
Add terms
0.077
SSR20.077<SSR10.2605
b Recall the residuals for the equations found in the previous part.
| x | Residual for y=-0.1x+10 | Residual for y=-0.05x+10 |
|---|---|---|
| 1 | -0.03 | -0.08 |
| 2 | 0.05 | -0.05 |
| 3 | -0.01 | -0.16 |
| 4 | 0.03 | -0.17 |
| 5 | 0.31 | 0.06 |
| 6 | 0.47 | 0.1 |
The points (x,residual) for each equation will be graphed on a scatter plot.
As can be seen, the residuals for Equation II are close the x-axis and therefore the sum of their squares has a lower value.
Example
Finding the Sum of Squared Residuals Less Than a Certain Value
Maya is researching used cars similar to the one her older sister drives. She found some data showing the mileages x, in thousands of miles, and the selling prices y, in thousands dollars, of several used cars near her.
| x | 23 | 12 | 18 | 30 | 6 | 26 |
|---|---|---|---|---|---|---|
| y | 15 | 15 | 17 | 12 | 19 | 15 |
Answer
Example Equation: y=-0.25x+20
Hint
Start by making a scatter plot for the given data set. Draw a line that passes through two data points and write its equation. Then, calculate the residuals for the equation.
Solution
First, the given data will be shown on a graph. Then, a line that passes close to the points will be drawn.
It appears that the line passes through the points (4,20) and (16,16). Knowing two points on the line, the slope of the line and its equation can be found. To find the slope, the points need to be substituted into the Slope Formula.
Next, using m and either point, the equation of the line can be written in the point-slope form. Use m=-31 and (x1,y1)=(4,20).
Finally, the residuals for this equation can be calculated. To do so, the difference between the y-value of a data point and the corresponding y-value predicted by the line of fit will be calculated for each data point.
Next, the sum of squared residuals can be found by adding the squares of the numbers in the last column of the table.
The sum of squared residuals for the equation is 16.
The residuals for this equation can be found as follows.
Now, the sum of squared residuals for this equation can be calculated.
As a result, the sum of squared residuals for y=-0.25x+20 satisfies the condition.
m=x2−x1y2−y1
SubstitutePoints
Substitute (4,20) & (16,16)
m=16−416−20
▼
Evaluate right-hand side
SubTerm
Subtract term
m=12-4
ReduceFrac
ba=b/4a/4
m=3-1
MoveNegNumToFrac
Put minus sign in front of fraction
m=-31
y−y1=m(x−x1)⇓y−20=-31(x−4)
To write the equation in slope-intercept form, y will be isolated.
y−20=-31(x−4)
▼
Solve for y
Distr
Distribute -31
y−20=-31x+34
AddEqn
LHS+20=RHS+20
y=-31x+34+20
NumberToFrac
a=33⋅a
y=-31x+34+360
AddFrac
Add fractions
y=-31x+364
| x | y (Actual) | y Predicted by the equation | Residual |
|---|---|---|---|
| 6 | 19 | y=-31(6)+364=358 | 19−358=-31 |
| 12 | 15 | y=-31(12)+364=352 | 15−352=-37 |
| 18 | 17 | y=-31(18)+364=346 | 17−346=35 |
| 23 | 15 | y=-31(23)+364=341 | 15−341=34 |
| 26 | 15 | y=-31(26)+364=338 | 15−358=37 |
| 30 | 12 | y=-31(30)+364=334 | 12−334=32 |
(-31)2+(-37)2+(35)2+(34)2+(37)2+(32)2
▼
Evaluate
NegBaseToPosBase
(-a)2=a2
(31)2+(37)2+(35)2+(34)2+(37)2+(32)2
PowQuot
(ba)m=bmam
3212+3272+3252+3242+3272+3222
CalcPow
Calculate power
91+949+925+916+949+94
AddFrac
Add fractions
9144
CalcQuot
Calculate quotient
16
Equationy=-31x+364Sum of Residuals16
This equation does not satisfy the condition that Maya wants. Another equation can be found by slightly increasing the slope of the above equation and slightly decreasing its y-intercept. Use the applet below to find an equation. For example, y=-0.25x+20 can be a good candidate. | x | y (Actual) | y Predicted by the equation | Residual |
|---|---|---|---|
| 6 | 19 | y=-0.25(6)+20=18.5 | 19−18.5=0.5 |
| 12 | 15 | y=-0.25(12)+20=17 | 15−17=-2 |
| 18 | 17 | y=-0.25(18)+20=15.5 | 17−15.5=1.5 |
| 23 | 15 | y=-0.25(23)+20=14.25 | 15−14.25=0.75 |
| 26 | 15 | y=-0.25(26)+20=13.5 | 15−13.5=1.5 |
| 30 | 12 | y=-0.25(30)+20=12.5 | 12−12.5=-0.5 |
(0.5)2+(-2)2+(1.5)2+(0.75)2+(1.5)2+(-0.5)2
▼
Evaluate
NegBaseToPosBase
(-a)2=a2
(0.5)2+(2)2+(1.5)2+(0.75)2+(1.5)2+(0.5)2
CalcPow
Calculate power
0.25+4+2.25+0.5625+2.25+0.25
AddTerms
Add terms
9.5625
Equationy=-31x+364y=-0.25x+20Sum of Residuals16>109.5625<10
Note that there are numerous equations that satisfy Maya's condition. Here only one of them was shown. The following applet shows the equation of a line fit and the sum of squared residuals. Use it to see how the sum changes as the equation changes. Pop Quiz
Practice Determining the Better Model
The applet generates bivariate data and two equations that model the data. To see the coordinates of a data point, move the cursor over it. Use the sum of squared residuals to determine the better fit for the data.
Closure
Finding the Line That Is the Best Fit
In this lesson, lines of fit and their residuals have been analyzed. When a line models a data set well, the sum of the squared residuals for the line is relatively small. Move the points and the line in the graph to see how the residuals and the sum of their squares change. 
The question then arises as to whether there is a line whose sum of squared residuals is less than the sum of the squared residuals of any other line of fit. In the next lesson, the line of best fit will be defined and its equation will be derived.
Lines of Fit
Exercises
The equation y=3x−4 models the data in the table.
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y | -12 | -7 | -5 | -1 | 0 | 4 | 6 |
Which graph is the scatter plot of the residuals?
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Let's make a table of the residual values.
| x | y | y Predicted by y=3x-4 | Residual |
|---|---|---|---|
| - 3 | - 12 | 3( - 3)-4= - 13 | - 12-( - 13)= 1 |
| - 2 | - 7 | 3( - 2)-4= - 10 | - 7-( - 10)= 3 |
| - 1 | - 5 | 3( - 1)-4= - 7 | - 5-( - 7)= 2 |
| 0 | - 1 | 3( 0)-4= - 4 | - 1-( - 4)= 3 |
| 1 | 0 | 3( 1)-4= - 1 | 0-( - 1)= 1 |
| 2 | 4 | 3( 2)-4= 2 | 4- 2= 2 |
| 3 | 6 | 3( 3)-4= 5 | 6- 5= 1 |
Now we can create a scatter plot using the given x-values and our residuals.
This graph corresponds to option C.
If the model is a good fit for the data, the scatter plot will be evenly distributed above and below the x-axis. In other words, there should not be any apparent patterns. Let's examine the residual plot.
As we can see, all residuals are positive and are not evenly distributed above and below the x-axis. We can conclude that the line of fit y=3x-4 does not model the given data set well.
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Mountain goats are exciting animals that can be seen in many parts of the world. The horns are an iconic characteristic of the male goat, also called a ram.
The table shows the length y, in centimeters, of a goat's horn each year. The equation y=2x+14.5 models the data in the table.
| Year, x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Growth, y | 15.2 | 18.5 | 21.4 | 22.7 | 23.8 |
Which graph is the scatter plot of the residuals?
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Let's make a table of the residual values.
| x | y | y Predicted by y=2x+14.5 | Residual |
|---|---|---|---|
| 1 | 15.2 | 2( 1)+14.5= 16.5 | 15.2- 16.5= - 1.3 |
| 2 | 18.5 | 2( 2)+14.5= 18.5 | 18.5- 18.5= 0 |
| 3 | 21.4 | 2( 3)+14.5= 20.5 | 21.4- 20.5= 0.9 |
| 4 | 22.7 | 2( 4)+14.5= 22.5 | 22.7- 22.5= 0.2 |
| 5 | 23.8 | 2( 5)+14.5= 24.5 | 23.8- 24.5= - 0.7 |
Now, we can create a scatter plot using the given x-values and our residuals.
This graph corresponds to option B.
We know that an equation is a good fit for the data, if the scatter plot is evenly distributed above and below the x-axis. Additionally, there should not be apparent patterns, which would indicate that a linear model is appropriate.
The line of fit does not model the data well because we see that a ⋂-shaped pattern appears.
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Paulina does some research about a profession she wants to have in the future. She records the experience and hourly wage for a sample of six people who work in that field.
| Experience, x | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Hourly Wage, y | 15.00 | 16.00 | 17.00 | 19.00 | 20.25 | 22.50 |
I.II.III.Linear model is not appropriate.The sum of the absolute value ofthe residuals is 1.75.The sum of squared residuals is 0.75.
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To find which statement(s) is correct, we need to find the residuals for Paulina's model.
Statement I
Let's make a table of the residual values.
| x | y | y Predicted by y=1.5x+13 | Residual |
|---|---|---|---|
| 1 | 15.00 | 1.5( 1)+13 = 14.5 | 15.00- 14.5= 0.5 |
| 2 | 16.00 | 1.5( 2)+13 = 16 | 16.00- 16= 0 |
| 3 | 17.00 | 1.5( 3)+13 = 17.5 | 17.00- 17.5= - 0.5 |
| 4 | 19.00 | 1.5( 4)+13 = 19 | 19.00- 19= 0 |
| 5 | 20.25 | 1.5( 5)+13 = 20.5 | 20.25- 20.5= - 0.25 |
| 6 | 22.50 | 1.5( 6)+13 = 22 | 22.50- 22= 0.5 |
Now we can create a scatter plot using the given x-values and our residuals.
We see that the scatter plot is evenly distributed above and below the x-axis. Additionally, there is no apparent pattern. Therefore, we can conclude that a linear model is appropriate. Statement I is false.
Statement II
Next, we will calculate the sum of the absolute value of the residuals.
It is equal to 1.75. Statment II is true.
Statement III
Finally, we will calculate the sum of squared residuals.
The sum of squared residuals is not equal to 0.75, which means that Statement III is also false. Therefore, only Statement II is true.
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Vincenzo is excited to be studying linear models.
He models two data sets using linear models. Consider the residual plots for each data set.
A.B.C.D.Linear models describe both data sets well.Linear model is appropriate for the first data set.Linear model describes the second data set well.Non-linear model is appropriate for both data sets.
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We determine the appropriate models by looking at the scatter plot of residuals. There are two things to consider.
- If the residuals are randomly placed about the x-axis, then a linear model describes the data set well.
- If some kind of pattern appears in the scatter plot, a non-linear model is more appropriate for the data.
Let's now consider the given residual plots.
We see that the residuals of Data Set I are evenly distributed above and below the x-axis and there is no apparent pattern. Therefore, a linear model is appropriate for Data Set I.
However, we can recognize a ⋃-shaped pattern in the residual plot for Data Set II, which means that a non-linear model is appropriate for Data Set II. Therefore, the statement in option B is true.
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Lines of Fit
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