single linear regression
DESCRIPTION
Single linear regressionTRANSCRIPT
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Single Linear Regression
Conceptual Explanation
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• Welcome to this explanation of Single Linear Regression.
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• Welcome to this explanation of Single Linear Regression.• Single linear regression is an extension of
correlation.
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• Welcome to this explanation of Single Linear Regression.• Single linear regression is an extension of
correlation.
Correlation Single Linear Regressionextends to
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• Correlation is designed to render a single coefficient that represents the degree of coherence between two variables
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• Correlation is designed to render a single coefficient that represents the degree of coherence between two variables
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• Correlation is designed to render a single coefficient that represents the degree of coherence between two variables
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• Correlation is designed to render a single coefficient that represents the degree of coherence between two variables
+.99
As one variable
increases the other
increases
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• Correlation is designed to render a single coefficient that represents the degree of coherence between two variables
+.99
As one variable
increases the other
increases
This coefficient represents an almost perfect positive
correlation or relationship between these two variables.
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• Correlation is designed to render a single coefficient that represents the degree of coherence between two variables
Ave Daily Temp
500
600
700
800
900
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• Correlation is designed to render a single coefficient that represents the degree of coherence between two variables
Ave Daily Temp
500
600
700
800
900
As one variable
decreases the other
increases
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• Correlation is designed to render a single coefficient that represents the degree of coherence between two variables
Ave Daily Temp
500
600
700
800
900
-.99
As one variable
decreases the other
increases
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• Correlation is designed to render a single coefficient that represents the degree of coherence between two variables
Ave Daily Temp
500
600
700
800
900
-.99
As one variable
decreases the other
increases
Almost a perfect negative correlation or relationship
between these two variables.
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• Single linear regression uses that information to predict the value of one variable based on the given value of the other variable.
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• Single linear regression uses that information to predict the value of one variable based on the given value of the other variable.
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• Single linear regression uses that information to predict the value of one variable based on the given value of the other variable.
• For example:
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• For example:If the following data set were real, what would you predict ice cream sales would be when the temperature reaches 1000?
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• For example:If the following data set were real, what would you predict ice cream sales would be when the temperature reaches 1000?
Ave Daily Ice Cream Sales
?560
480
350
320
230
Ave Daily Temp
1000
900
800
700
600
500
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• Single linear regression uses that information to predict the value of one variable (ice cream) based on the given value of the other variable (temperature).
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• Single linear regression uses that information to predict the value of one variable (ice cream) based on the given value of the other variable (temperature).
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If the following data set were real, what would you predict ice cream sales would be when the temperature reaches 1000?
• Rather than simply examining the relationship between the variables (as is the case with the Pearson Product Moment Correlation), one variable will be used as the predictor (temperature) and the other value will be used as the outcome or predicted (ice cream sales).
Ave Daily Ice Cream Sales
630?560
480
350
320
230
Ave Daily Temp
1000
900
800
700
600
500
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If the following data set were real, what would you predict ice cream sales would be when the temperature reaches 1000?
• Rather than simply examining the relationship between the variables (as is the case with the Pearson Product Moment Correlation), one variable will be used as the predictor (temperature) and the other value will be used as the outcome or predicted (ice cream sales). • Linear Regression makes it possible to estimate a value like
630
Ave Daily Ice Cream Sales
630?560
480
350
320
230
Ave Daily Temp
1000
900
800
700
600
500
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• In some cases which variable is considered predictor or outcome is arbitrary.
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• In some cases which variable is considered predictor or outcome is arbitrary.• Like measures of depression and anxiety
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• In some cases which variable is considered predictor or outcome is arbitrary.• Like measures of depression and anxiety
Composite Depression Score
33
26
22
14
12
6
Composite Anxiety Score
103
100
92
74
52
26
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• In some cases which variable is considered predictor or outcome is arbitrary.• Like measures of depression and anxiety
• It’s not clear which influences which. Most likely depression and anxiety mutually influence one another.
Composite Depression Score
33
26
22
14
12
6
Composite Anxiety Score
103
100
92
74
52
26
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• In some cases, either by theory or by the nature of the research design, one variable will be rationally defined as the predictor and the other as the outcome.
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• In some cases, either by theory or by the nature of the research design, one variable will be rationally defined as the predictor and the other as the outcome.
Ave Daily Exposure to Sunlight
3.3 hrs
2.6 hrs
2.2 hrs
1.4 hrs
1.2 hrs
0.6 hrs
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• In some cases, either by theory or by the nature of the research design, one variable will be rationally defined as the predictor and the other as the outcome.
Ave Daily Exposure to Sunlight
3.3 hrs
2.6 hrs
2.2 hrs
1.4 hrs
1.2 hrs
0.6 hrs
Levels of Vitamin E after two months
10.3 units
8.1 units
7.3 units
7.0 units
6.8 units
5.7 units
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• In some cases, either by theory or by the nature of the research design, one variable will be rationally defined as the predictor and the other as the outcome.
Ave Daily Exposure to Sunlight
3.3 hrs
2.6 hrs
2.2 hrs
1.4 hrs
1.2 hrs
0.6 hrs
Levels of Vitamin E after two months
10.3 units
8.1 units
7.3 units
7.0 units
6.8 units
5.7 units
In this example, exposure to sunlight may impact levels of
Vitamin E.
But, levels of Vitamin E would not impact the amount of sunlight
one gets.
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• An easy way to conceptualize single linear regression is to create a scatterplot in Cartesian space.
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• An easy way to conceptualize single linear regression is to create a scatterplot in Cartesian space.
Let’s plot the following data set:
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• An easy way to conceptualize single linear regression is to create a scatterplot in Cartesian space.
Let’s plot the following data set:Composite
Depression Score33
26
22
14
12
6
Composite Anxiety Score
103
100
92
74
52
26
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• First, we assign the predictor variable along the X axis, which in this case we’ll arbitrarily say is depression.
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• First, we assign the predictor variable along the X axis, which in this case we’ll arbitrarily say is depression.
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
Depression
Anx
iety
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• ... and the outcome variable along the Y axis we’ll arbitrarily say is Anxiety.
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• ... and the outcome variable along the Y axis we’ll arbitrarily say is Anxiety.
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
Depression
Anx
iety
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• Now, let’s identify or plot each point or dot
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• Now, let’s identify or plot each point or dotDepression
33262214126
Anxiety
10310092745226
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• Now, let’s identify or plot each point or dotDepression
33262214126
Anxiety
10310092745226
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
Depression
Anx
iety
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• Now, let’s identify or plot each point or dotDepression
33262214126
Anxiety
10310092745226
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
Depression
Anx
iety
(33, 103)
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• Now, let’s identify or plot each point or dotDepression
33262214126
Anxiety
10310092745226
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• Now, let’s identify or plot each point or dotDepression
33262214126
Anxiety
10310092745226
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
Depression
Anx
iety
(26, 100)
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• Now, let’s identify or plot each point or dotDepression
33262214126
Anxiety
10310092745226
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• Now, let’s identify or plot each point or dotDepression
33262214126
Anxiety
10310092745226
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
Depression
Anx
iety
(22, 92)
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• Now, let’s identify or plot each point or dotDepression
33262214126
Anxiety
10310092745226
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• Now, let’s identify or plot each point or dotDepression
33262214126
Anxiety
10310092745226
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
Depression
Anx
iety
(14, 74)
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• Now, let’s identify or plot each point or dotDepression
33262214126
Anxiety
10310092745226
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• Now, let’s identify or plot each point or dotDepression
33262214126
Anxiety
10310092745226
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
Depression
Anx
iety
(12, 52)
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• Now, let’s identify or plot each point or dotDepression
33262214126
Anxiety
10310092745226
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• Now, let’s identify or plot each point or dotDepression
33262214126
Anxiety
10310092745226
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
Depression
Anx
iety
(6, 26)
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• Visually, one can see in the plotted space whether there is a tendency for the variables to be related and in what direction they are related.
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• Visually, one can see in the plotted space whether there is a tendency for the variables to be related and in what direction they are related.
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
Depression
Anx
iety
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• Visually, one can see in the plotted space whether there is a tendency for the variables to be related and in what direction they are related.
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
Depression
Anx
iety
In this case there is a strong tendency
to relate and the relationship is
positive
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• With this data set the tendency for the variables to relate is strong and the direction is negative:
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• With this data set the tendency for the variables to relate is strong and the direction is negative:
Depression
61214222633
Anxiety
10310092745226
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• With this data set the tendency for the variables to relate is strong and the direction is negative:
Depression
61214222633
Anxiety
10310092745226
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
Depression
Anx
iety
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• With this data set the tendency for the variables to relate is strong and the direction is negative:
Depression
61214222633
Anxiety
10310092745226
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
Depression
Anx
iety
Strong and Negative
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• When no relationship exists the scatter plot tends to look like a big circle.
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• When no relationship exists the scatter plot tends to look like a big circle.
Depression
2233126
1426
Anxiety
10310092745226
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• When no relationship exists the scatter plot tends to look like a big circle.
Depression
2233126
1426
Anxiety
10310092745226
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
DepressionA
nxie
ty
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• When no relationship exists the scatter plot tends to look like a big circle.
Depression
2233126
1426
Anxiety
10310092745226
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
DepressionA
nxie
ty
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• When no relationship exists the scatter plot tends to look like a big circle.
Depression
226
33261412
Anxiety
10310092745226
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0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
DepressionA
nxie
ty
• When no relationship exists the scatter plot tends to look like a big circle.
Depression
226
33261412
Anxiety
10310092745226
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0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
DepressionA
nxie
ty
• When no relationship exists the scatter plot tends to look like a big circle.
Depression
226
33261412
Anxiety
10310092745226
Weak and Positive
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• When no relationship exists the scatter plot tends to look like a big circle.
Depression
61433261222
Anxiety
10310074925226
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0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
DepressionA
nxie
ty
• When no relationship exists the scatter plot tends to look like a big circle.
Depression
61433261222
Anxiety
10310074925226
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0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
DepressionA
nxie
ty
• When no relationship exists the scatter plot tends to look like a big circle.
Depression
61433261222
Anxiety
10310074925226
Weak and Negative
![Page 69: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/69.jpg)
• You might have noticed that as the variables are related either positively or negatively, the plot looks more like an oval tilted one way or the other.
![Page 70: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/70.jpg)
• You might have noticed that as the variables are related either positively or negatively, the plot looks more like an oval tilted one way or the other.
0 5 10 15 20 25 30 350
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Relationship between Depression & Anxiety
Depression
Anx
iety
0 5 10 15 20 25 30 350
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Relationship between Depression & Anxiety
DepressionA
nxie
ty
![Page 71: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/71.jpg)
• You might have noticed that as the variables are related either positively or negatively, the plot looks more like an oval tilted one way or the other.
Weak and Negative
0 5 10 15 20 25 30 350
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40
60
80
100
120
Relationship between Depression & Anxiety
Depression
Anx
iety
0 5 10 15 20 25 30 350
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60
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Relationship between Depression & Anxiety
DepressionA
nxie
ty
Weak and Positive
![Page 72: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/72.jpg)
• As mentioned before, Linear Regression is used to predict one variable (ice cream sales) from another related variable (temperature).
![Page 73: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/73.jpg)
• As mentioned before, Linear Regression is used to predict one variable (ice cream sales) from another related variable (temperature). • The stronger the relationship (e.g., +.99 or -.99) the more
accurate the prediction.
![Page 74: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/74.jpg)
• As mentioned before, Linear Regression is used to predict one variable (ice cream sales) from another related variable (temperature). • The stronger the relationship (e.g., +.99 or -.99) the more
accurate the prediction.
• The weaker the relationship (e.g., +.14 or -.03) the less accurate the prediction.
![Page 75: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/75.jpg)
• As mentioned before, Linear Regression is used to predict one variable (ice cream sales) from another related variable (temperature). • The stronger the relationship (e.g., +.99 or -.99) the more
accurate the prediction.
• The weaker the relationship (e.g., +.14 or -.03) the less accurate the prediction.
![Page 76: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/76.jpg)
• As mentioned before, Linear Regression is used to predict one variable (ice cream sales) from another related variable (temperature). • The stronger the relationship (e.g., +.99 or -.99) the more
accurate the prediction.
• The weaker the relationship (e.g., +.14 or -.03) the less accurate the prediction.
• One of the ways to represent those relationships is of course with the coefficients (e.g., +.99, +.14, -.03, -.99).
![Page 77: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/77.jpg)
• As mentioned before, Linear Regression is used to predict one variable (ice cream sales) from another related variable (temperature). • The stronger the relationship (e.g., +.99 or -.99) the more
accurate the prediction.
• The weaker the relationship (e.g., +.14 or -.03) the less accurate the prediction.
• One of the ways to represent those relationships is of course with the coefficients (e.g., +.99, +.14, -.03, -.99).• Another way to represent it is by graphing the relationship.
![Page 78: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/78.jpg)
• Recall that a line in Cartesian space is defined by its slope and its Y intercept (the value of Y when X equals 0).
![Page 79: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/79.jpg)
• Recall that a line in Cartesian space is defined by its slope and its Y intercept (the value of Y when X equals 0).
[Y= intercept + (slope X)]∙
![Page 80: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/80.jpg)
• Recall that a line in Cartesian space is defined by its slope and its Y intercept (the value of Y when X equals 0).
[Y= intercept + (slope X)]∙
0 1 2 3 4 5 60
1
2
3
4
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6
![Page 81: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/81.jpg)
• In this case the slope would be 1. You may remember that this value is derived by taking what is called the “rise” over the “run”.
![Page 82: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/82.jpg)
0 1 2 3 4 5 60
1
2
3
4
5
6
rise1
• In this case the slope would be 1. You may remember that this value is derived by taking what is called the “rise” over the “run”.
run1
![Page 83: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/83.jpg)
0 1 2 3 4 5 60
1
2
3
4
5
6
rise1
• In this case the slope would be 1. You may remember that this value is derived by taking what is called the “rise” over the “run”.
• So the equation for this line so far would look like this:
run1
![Page 84: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/84.jpg)
0 1 2 3 4 5 60
1
2
3
4
5
6
rise1
• In this case the slope would be 1. You may remember that this value is derived by taking what is called the “rise” over the “run”.
• So the equation for this line so far would look like this:
run1
𝒚=0+11𝒙
![Page 85: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/85.jpg)
0 1 2 3 4 5 60
1
2
3
4
5
6
rise1
run1
𝒚=0+11𝒙
![Page 86: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/86.jpg)
0 1 2 3 4 5 60
1
2
3
4
5
6
rise1
run1
𝒚=0+11𝒙
This is where the line crosses the
Y axis.
![Page 87: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/87.jpg)
0 1 2 3 4 5 60
1
2
3
4
5
6
rise1
run1
𝒚=0+11𝒙
This is the slope which is the rise
over the run.
![Page 88: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/88.jpg)
• A line represents the functional relationship between variable X and variable Y, therefore, that line can be used to predict a Y value from any given X value.
![Page 89: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/89.jpg)
• A line represents the functional relationship between variable X and variable Y, therefore, that line can be used to predict a Y value from any given X value.
Feb
Mar
Apr
May
Jun
Ave Monthly Temperature
500
600
700
800
900
Ave Monthly Ice Cream Sales
239
320
400
480
560
![Page 90: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/90.jpg)
• In this case the two variables (temperature and ice cream sales) have a perfect linear relationship. This is rarely ever seen among variables such as these in the real world, but for illustrative purposes we have created a perfect relationship.
![Page 91: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/91.jpg)
• In this case the two variables (temperature and ice cream sales) have a perfect linear relationship. This is rarely ever seen among variables such as these in the real world, but for illustrative purposes we have created a perfect relationship.
40 60 80 100 1200
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200
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400
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600
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Average Monthly Ice Cream Sales
Ave
Mon
thly
Tem
pera
ture
![Page 92: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/92.jpg)
• Now let’s say we have data for the average temperature during the month of July. But, we don’t have the data for the average ice cream sales for July
![Page 93: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/93.jpg)
• Now let’s say we have data for the average temperature during the month of July. But, we don’t have the data for the average ice cream sales for July
Feb
Mar
Apr
May
Jun
JUL
Ave Monthly Temperature
500
600
700
800
900
1000
Ave Monthly Ice Cream Sales
239
320
400
480
560
?
![Page 94: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/94.jpg)
• Now let’s say we have data for the average temperature during the month of July. But, we don’t have the data for the average ice cream sales for July
• Using single linear regression we can predict the average ice cream sales for July. Here is the formula we will use for the prediction:
Feb
Mar
Apr
May
Jun
JUL
Ave Monthly Temperature
500
600
700
800
900
1000
Ave Monthly Ice Cream Sales
239
320
400
480
560
?
![Page 95: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/95.jpg)
• Now let’s say we have data for the average temperature during the month of July. But, we don’t have the data for the average ice cream sales for July
• Using single linear regression we can predict the average ice cream sales for July. Here is the formula we will use for the prediction:
Feb
Mar
Apr
May
Jun
JUL
Ave Monthly Temperature
500
600
700
800
900
1000
Ave Monthly Ice Cream Sales
239
320
400
480
560
?
¿
![Page 96: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/96.jpg)
• Now let’s say we have data for the average temperature during the month of July. But, we don’t have the data for the average ice cream sales for July
• Using single linear regression we can predict the average ice cream sales for July. Here is the formula we will use for the prediction:
• There are many ways to write this equation. Here is one way:
Feb
Mar
Apr
May
Jun
JUL
Ave Monthly Temperature
500
600
700
800
900
1000
Ave Monthly Ice Cream Sales
239
320
400
480
560
?
¿
![Page 97: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/97.jpg)
• Now let’s say we have data for the average temperature during the month of July. But, we don’t have the data for the average ice cream sales for July
• Using single linear regression we can predict the average ice cream sales for July. Here is the formula we will use for the prediction:
• There are many ways to write this equation. Here is one way:
Feb
Mar
Apr
May
Jun
JUL
Ave Monthly Temperature
500
600
700
800
900
1000
Ave Monthly Ice Cream Sales
239
320
400
480
560
?
¿
¿
![Page 98: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/98.jpg)
• Using this data set we can create a formula for a straight line that represents that relationship:
![Page 99: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/99.jpg)
• Using this data set we can create a formula for a straight line that represents that relationship:
Feb
Mar
Apr
May
Jun
Ave Monthly Temperature
500
600
700
800
900
Ave Monthly Ice Cream Sales
239
320
400
480
560
![Page 100: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/100.jpg)
• Using this data set we can create a formula for a straight line that represents that relationship:
Feb
Mar
Apr
May
Jun
Ave Monthly Temperature
500
600
700
800
900
Ave Monthly Ice Cream Sales
239
320
400
480
560
40 60 80 100 1200
100
200
300
400
500
600
700
Average Monthly Ice Cream Sales
Ave
Mon
thly
Tem
pera
ture
𝑦= -162+8( )𝑥
![Page 101: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/101.jpg)
• With this equation we can now plug in the average temperature for July (1000) and see what the predicted average ice cream sales would be:
![Page 102: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/102.jpg)
• With this equation we can now plug in the average temperature for July (1000) and see what the predicted average ice cream sales would be:
Feb
Mar
Apr
May
Jun
Jul
Ave Monthly Temperature
500
600
700
800
900
1000
Ave Monthly Ice Cream Sales
239
320
400
480
560
![Page 103: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/103.jpg)
• With this equation we can now plug in the average temperature for July (1000) and see what the predicted average ice cream sales would be:
Feb
Mar
Apr
May
Jun
Jul
Ave Monthly Temperature
500
600
700
800
900
1000
Ave Monthly Ice Cream Sales
239
320
400
480
560
40 60 80 100 1200
100
200
300
400
500
600
700
Average Monthly Ice Cream Sales
Ave
Mon
thly
Tem
pera
ture
𝑦 ̂� = -162 + 8(100)
![Page 104: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/104.jpg)
• With this equation we can now plug in the average temperature for July (1000) and see what the predicted average ice cream sales would be:
Feb
Mar
Apr
May
Jun
Jul
Ave Monthly Temperature
500
600
700
800
900
1000
Ave Monthly Ice Cream Sales
239
320
400
480
560
40 60 80 100 1200
100
200
300
400
500
600
700
Average Monthly Ice Cream Sales
Ave
Mon
thly
Tem
pera
ture
𝑦 ̂� = -162 + 8(100)
![Page 105: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/105.jpg)
• With this equation we can now plug in the average temperature for July (1000) and see what the predicted average ice cream sales would be:
Feb
Mar
Apr
May
Jun
Jul
Ave Monthly Temperature
500
600
700
800
900
1000
Ave Monthly Ice Cream Sales
239
320
400
480
560
40 60 80 100 1200
100
200
300
400
500
600
700
Average Monthly Ice Cream Sales
Ave
Mon
thly
Tem
pera
ture
𝑦 ̂� = -162 + 800
![Page 106: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/106.jpg)
• With this equation we can now plug in the average temperature for July (1000) and see what the predicted average ice cream sales would be:
Feb
Mar
Apr
May
Jun
Jul
Ave Monthly Temperature
500
600
700
800
900
1000
Ave Monthly Ice Cream Sales
239
320
400
480
560
40 60 80 100 1200
100
200
300
400
500
600
700
Average Monthly Ice Cream Sales
Ave
Mon
thly
Tem
pera
ture
𝑦 ̂� = 638
![Page 107: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/107.jpg)
• With this equation we can now plug in the average temperature for July (1000) and see what the predicted average ice cream sales would be:
Feb
Mar
Apr
May
Jun
Jul
Ave Monthly Temperature
500
600
700
800
900
1000
Ave Monthly Ice Cream Sales
239
320
400
480
560
638
40 60 80 100 1200
100
200
300
400
500
600
700
Average Monthly Ice Cream Sales
Ave
Mon
thly
Tem
pera
ture
𝑦 ̂� = 638
![Page 108: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/108.jpg)
• So, based on our single linear regression analysis we would predict that in the month of July that the average monthly ice cream sales will be 638.
Feb
Mar
Apr
May
Jun
Jul
Ave Monthly Temperature
500
600
700
800
900
1000
Ave Monthly Ice Cream Sales
239
320
400
480
560
638
40 60 80 100 1200
100
200
300
400
500
600
700
Average Monthly Ice Cream Sales
Ave
Mon
thly
Tem
pera
ture
𝑦 ̂� = 638
![Page 109: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/109.jpg)
• So, based on our single linear regression analysis we would predict that in the month of July that the average monthly ice cream sales will be 638.
• This is a simple demonstration of how regression works.
Feb
Mar
Apr
May
Jun
Jul
Ave Monthly Temperature
500
600
700
800
900
1000
Ave Monthly Ice Cream Sales
239
320
400
480
560
638
40 60 80 100 1200
100
200
300
400
500
600
700
Average Monthly Ice Cream Sales
Ave
Mon
thly
Tem
pera
ture
𝑦 ̂� = 638
![Page 110: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/110.jpg)
• So, based on our single linear regression analysis we would predict that in the month of July that the average monthly ice cream sales will be 638.
• This is a simple demonstration of how regression works.• In reality, however, most variables will not correlate so
perfectly like this did:
Feb
Mar
Apr
May
Jun
Jul
Ave Monthly Temperature
500
600
700
800
900
1000
Ave Monthly Ice Cream Sales
239
320
400
480
560
638
40 60 80 100 1200
100
200
300
400
500
600
700
Average Monthly Ice Cream Sales
Ave
Mon
thly
Tem
pera
ture
𝑦 ̂� = 638
![Page 111: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/111.jpg)
• Most will look like this:
![Page 112: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/112.jpg)
• Most will look like this:
![Page 113: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/113.jpg)
• Most will look like this:
• This line is called the best fitting line because it minimizes the distance between the line and all of the points. You will notice again that we have a linear equation for that line:
![Page 114: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/114.jpg)
• Most will look like this:
• This line is called the best fitting line because it minimizes the distance between the line and all of the points. You will notice again that we have a linear equation for that line:𝑦= -
50.93+7.21(x)
![Page 115: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/115.jpg)
• Most will look like this:
• This equation is calculated by using the standard deviations and means of the two variables. For brevity sake we will not go into this here. 𝑦= -
50.93+7.21(x)
![Page 116: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/116.jpg)
• Given the infinite number of positive linear fitting through a scatterplot, the one closer to represent the functional relationship between X and Y is the line that results in the cumulative least squared error between the predicted values of Y and the true observed values of Y for each given X.
![Page 117: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/117.jpg)
• Given the infinite number of positive linear fitting through a scatterplot, the one closer to represent the functional relationship between X and Y is the line that results in the cumulative least squared error between the predicted values of Y and the true observed values of Y for each given X.
![Page 118: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/118.jpg)
• Given the infinite number of positive linear fitting through a scatterplot, the one closer to represent the functional relationship between X and Y is the line that results in the cumulative least squared error between the predicted values of Y and the true observed values of Y for each given X.
This line is the predicted values of Y calculated from the
equation
![Page 119: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/119.jpg)
• Given the infinite number of positive linear fitting through a scatterplot, the one closer to represent the functional relationship between X and Y is the line that results in the cumulative least squared error between the predicted values of Y and the true observed values of Y for each given X.
These dots represent the actual data
This line is the predicted values of Y calculated from the
equation
![Page 120: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/120.jpg)
• We don’t have to actually plot the coordinates and lines. We can operate solely on the equations to generate predicted values and errors in prediction. In this way we can determine if temperature is a statistically significant predictor of ice cream sales.
![Page 121: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/121.jpg)
• So here are the actual data we plotted the data from:
![Page 122: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/122.jpg)
• So here are the actual data we plotted the data from:
Jan 40 300
Feb 50 320
Mar 60 370
Apr 70 480
May 80 560
Jun 90 640
Jul 100 720
Aug 90 600
Sep 80 400
Oct 60 300
Nov 40 200
Dec 20 122
(X) Ave Monthly
Temp
(y) Actual Ave Monthly Ice Cream Sales
![Page 123: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/123.jpg)
• So here are the actual data we plotted the data from:
Jan 40 300
Feb 50 320
Mar 60 370
Apr 70 480
May 80 560
Jun 90 640
Jul 100 720
Aug 90 600
Sep 80 400
Oct 60 300
Nov 40 200
Dec 20 122
(X) Ave Monthly
Temp
(y) Actual Ave Monthly Ice Cream Sales
![Page 124: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/124.jpg)
• So here are the actual data we plotted the data from:
Jan 40 300
Feb 50 320
Mar 60 370
Apr 70 480
May 80 560
Jun 90 640
Jul 100 720
Aug 90 600
Sep 80 400
Oct 60 300
Nov 40 200
Dec 20 122
(X) Ave Monthly
Temp
(y) Actual Ave Monthly Ice Cream Sales
• We can now plot the predicted Y using the equation:
![Page 125: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/125.jpg)
• So here are the actual data we plotted the data from:
Jan 40 300
Feb 50 320
Mar 60 370
Apr 70 480
May 80 560
Jun 90 640
Jul 100 720
Aug 90 600
Sep 80 400
Oct 60 300
Nov 40 200
Dec 20 122
(X) Ave Monthly
Temp
(y) Actual Ave Monthly Ice Cream Sales
• We can now plot the predicted Y using the equation: = -50.93+7.21(x)
![Page 126: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/126.jpg)
• So here are the actual data we plotted the data from:
Jan 40 300
Feb 50 320
Mar 60 370
Apr 70 480
May 80 560
Jun 90 640
Jul 100 720
Aug 90 600
Sep 80 400
Oct 60 300
Nov 40 200
Dec 20 122
(X) Ave Monthly
Temp
(y) Actual Ave Monthly Ice Cream Sales
• We can now plot the predicted Y using the equation:
• Which is the equation for the best fitting line between these two variables:
= -50.93+7.21(x)
![Page 127: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/127.jpg)
• We can now plot the predicted Y using the equation:
![Page 128: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/128.jpg)
• We can now plot the predicted Y using the equation:= -50.93+7.21(x)
![Page 129: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/129.jpg)
• We can now plot the predicted Y using the equation:
Jan 40 300
Feb 50 320
Mar 60 370
Apr 70 480
May 80 560
Jun 90 640
Jul 100 720
Aug 90 600
Sep 80 400
Oct 60 300
Nov 40 200
Dec 20 122
(X) Ave Monthly
Temp
(y) Actual Ave Monthly Ice Cream Sales
= -50.93+7.21(x)
![Page 130: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/130.jpg)
• We can now plot the predicted Y using the equation:
Jan 40 300
Feb 50 320
Mar 60 370
Apr 70 480
May 80 560
Jun 90 640
Jul 100 720
Aug 90 600
Sep 80 400
Oct 60 300
Nov 40 200
Dec 20 122
(X) Ave Monthly
Temp
(y) Actual Ave Monthly Ice Cream Sales
= -50.93+7.21(x)
= -50.93+7.21(300) == -50.93+7.21(320) =
= -50.93+7.21(480) =
= -50.93+7.21(370) =
= -50.93+7.21(560) =
= -50.93+7.21(640) =
= -50.93+7.21(720) =
= -50.93+7.21(600) =
= -50.93+7.21(400) == -50.93+7.21(300) =
= -50.93+7.21(200) == -50.93+7.21(122) =
![Page 131: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/131.jpg)
• We can now plot the predicted Y using the equation:
Jan 40 300
Feb 50 320
Mar 60 370
Apr 70 480
May 80 560
Jun 90 640
Jul 100 720
Aug 90 600
Sep 80 400
Oct 60 300
Nov 40 200
Dec 20 122
(X) Ave Monthly
Temp
(y) Actual Ave Monthly Ice Cream Sales
= -50.93+7.21(x)
= -50.93+7.21(300) == -50.93+7.21(320) =
= -50.93+7.21(480) =
= -50.93+7.21(370) =
= -50.93+7.21(560) =
= -50.93+7.21(640) =
= -50.93+7.21(720) =
= -50.93+7.21(600) =
= -50.93+7.21(400) == -50.93+7.21(300) =
= -50.93+7.21(200) == -50.93+7.21(122) =
Predicted Ave Monthly Ice Cream
Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
![Page 132: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/132.jpg)
• We can now plot the predicted Y using the equation:
• With this information we can now determine if x (temperature) is a statistically significant predictor of “y” (ice cream sales).
Jan 40 300
Feb 50 320
Mar 60 370
Apr 70 480
May 80 560
Jun 90 640
Jul 100 720
Aug 90 600
Sep 80 400
Oct 60 300
Nov 40 200
Dec 20 122
(X) Ave Monthly
Temp
(y) Actual Ave Monthly Ice Cream Sales
= -50.93+7.21(x)
= -50.93+7.21(300) == -50.93+7.21(320) =
= -50.93+7.21(480) =
= -50.93+7.21(370) =
= -50.93+7.21(560) =
= -50.93+7.21(640) =
= -50.93+7.21(720) =
= -50.93+7.21(600) =
= -50.93+7.21(400) == -50.93+7.21(300) =
= -50.93+7.21(200) == -50.93+7.21(122) =
Predicted Ave Monthly Ice Cream
Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
![Page 133: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/133.jpg)
• To begin we need to determine the total sum of squares just like we would do with analysis of variance.
![Page 134: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/134.jpg)
• To begin we need to determine the total sum of squares just like we would do with analysis of variance.
• This is done by subtracting the actual “Y” (ice cream sales) values from the average or mean ice cream sales for the whole year.
![Page 135: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/135.jpg)
• To begin we need to determine the total sum of squares just like we would do with analysis of variance.
• This is done by subtracting the actual “Y” (ice cream sales) values from the average or mean ice cream sales for the whole year.
• The mean is calculated by adding up the values and divided them by how many there are.
![Page 136: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/136.jpg)
• To begin we need to determine the total sum of squares just like we would do with analysis of variance.
• This is done by subtracting the actual “Y” (ice cream sales) values from the average or mean ice cream sales for the whole year.
• The mean is calculated by adding up the values and divided them by how many there are.
• (300+320+370+480+560+640+720+600+400+300+200+122) / 12 = 417 average ice cream sales
![Page 137: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/137.jpg)
• We then subtract each y value from the mean
![Page 138: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/138.jpg)
• We then subtract each y value from the mean
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
Difference
-117
-97
-47
63
143
223
303
183
-17
-117
-217
-295
------------
============
![Page 139: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/139.jpg)
• We then subtract each y value from the mean
• Note - if we did not know the functional relationship between X and Y, our best prediction of any one person’s Y value would be the mean of Y.
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
Difference
-117
-97
-47
63
143
223
303
183
-17
-117
-217
-295
------------
============
![Page 140: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/140.jpg)
• Because we are calculating the total sum of squares we will need to square the results and then take the average of the sum of squares. This is the same as the variance of all of the scores.
![Page 141: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/141.jpg)
• Because we are calculating the total sum of squares we will need to square the results and then take the average of the sum of squares. This is the same as the variance of all of the scores.
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
Difference
-117
-97
-47
63
143
223
303
183
-17
-117
-217
-295
------------
============
Squared
13689
9409
2209
3969
20449
49729
91809
33489
289
13689
47089
87025
![Page 142: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/142.jpg)
• Because we are calculating the total sum of squares we will need to square the results and then sum up the results
![Page 143: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/143.jpg)
• Because we are calculating the total sum of squares we will need to square the results and then sum up the result
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
Difference
-117
-97
-47
63
143
223
303
183
-17
-117
-217
-295
------------
============
Squared
13689
9409
2209
3969
20449
49729
91809
33489
289
13689
47089
87025
Sum up
SUM 372844
![Page 144: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/144.jpg)
• Now we find regression (good) and residual (bad). To have better prediction power we want the regression sums of squares to be large and the residual or error sums of squares to be small.
![Page 145: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/145.jpg)
• Now we find regression (good) and residual (bad). To have better prediction power we want the regression sums of squares to be large and the residual or error sums of squares to be small.• Let’s see if the residual or the regression is greater.
![Page 146: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/146.jpg)
• Now we find regression (good) and residual (bad). To have better prediction power we want the regression sums of squares to be large and the residual or error sums of squares to be small.• Let’s see if the residual or the regression is greater.• We know that the total sums of squares is 31,070.
![Page 147: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/147.jpg)
• Now we find regression (good) and residual (bad). To have better prediction power we want the regression sums of squares to be large and the residual or error sums of squares to be small.• Let’s see if the residual or the regression is greater.• We know that the total sums of squares is 31,070.
Sum of Squares df Mean Square F-ratio Significance Total 372,844
![Page 148: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/148.jpg)
• Now we find regression (good) and residual (bad). To have better prediction power we want the regression sums of squares to be large and the residual or error sums of squares to be small.• Let’s see if the residual or the regression is greater.• We know that the total sums of squares is 31,070. • Now we will calculate the residual (error) and the
regression sums of squares which will add up to 372,844. Sum of Squares df Mean Square F-ratio Significance Total 372,844
![Page 149: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/149.jpg)
• Now we find regression (good) and residual (bad). To have better prediction power we want the regression sums of squares to be large and the residual or error sums of squares to be small.• Let’s see if the residual or the regression is greater.• We know that the total sums of squares is 31,070. • Now we will calculate the residual (error) and the
regression sums of squares which will add up to 372,844. Sum of Squares df Mean Square F-ratio SignificanceRegression ? Residual (error) ? Total 372,844
![Page 150: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/150.jpg)
• Before we calculate residual and regression let’s see visually how we calculated the total sums of squares -372,844.
![Page 151: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/151.jpg)
• Before we calculate residual and regression let’s see visually how we calculated the total sums of squares -372,844.• Once again we subtract the actual Y values from the mean
of the actual Y values
![Page 152: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/152.jpg)
• Before we calculate residual and regression let’s see visually how we calculated the total sums of squares -372,844.• Once again we subtract the actual Y values from the mean
of the actual Y values(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
417
417
417
417
417
417
417
417
417
417
417
417
------------
![Page 153: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/153.jpg)
• Before we calculate residual and regression let’s see visually how we calculated the total sums of squares -372,844.• Once again we subtract the actual Y values from the mean
of the actual Y values(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
417
417
417
417
417
417
417
417
417
417
417
417
------------
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
![Page 154: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/154.jpg)
• Before we calculate residual and regression let’s see visually how we calculated the total sums of squares -372,844.• Once again we subtract the actual Y values from the mean
of the actual Y values(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
417
417
417
417
417
417
417
417
417
417
417
417
------------
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
![Page 155: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/155.jpg)
• Before we calculate residual and regression let’s see visually how we calculated the total sums of squares -372,844.• Once again we subtract the actual Y values from the mean
of the actual Y values(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
417
417
417
417
417
417
417
417
417
417
417
417
------------
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
![Page 156: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/156.jpg)
• Before we calculate residual and regression let’s see visually how we calculated the total sums of squares -372,844.• Once again we subtract the actual Y values from the mean
of the actual Y values(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
417
417
417
417
417
417
417
417
417
417
417
417
------------
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
![Page 157: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/157.jpg)
• The first data set are the actual Y values. We subtract them from the mean (417) which would be our best prediction if we did not know the relationship between X (temperature) and Y (ice cream sales)
![Page 158: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/158.jpg)
• The first data set are the actual Y values. We subtract them from the mean (417) which would be our best prediction if we did not know the relationship between X (temperature) and Y (ice cream sales)
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
417
417
417
417
417
417
417
417
417
417
417
417
------------
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
![Page 159: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/159.jpg)
• Here is the graphic depiction of our subtracting each data point from the mean (417):
![Page 160: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/160.jpg)
• Here is the graphic depiction of our subtracting each data point from the mean (417):
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
122
Final Exam
Mid
term
Exa
m
122-417
= -295
417
![Page 161: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/161.jpg)
• Here is the graphic depiction of our subtracting each data point from the mean (417):
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
122
Final Exam
Mid
term
Exa
m
122-417
= -295
417
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
417
417
417
417
417
417
417
417
417
417
417
417
Difference
-117
-97
-47
63
143
223
303
183
-17
-117
-217
-295
------------
============
![Page 162: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/162.jpg)
• Here is the graphic depiction of our subtracting each data point from the mean (417):
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
122
Final Exam
Mid
term
Exa
m
122-417
= -295
417
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
417
417
417
417
417
417
417
417
417
417
417
417
Difference
-117
-97
-47
63
143
223
303
183
-17
-117
-217
-295
------------
============
![Page 163: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/163.jpg)
• Here is the graphic depiction of our subtracting each data point from the mean (417):
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
200
Final Exam
Mid
term
Exa
m
200-417
= -217
417
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
417
417
417
417
417
417
417
417
417
417
417
417
Difference
-117
-97
-47
63
143
223
303
183
-17
-117
-217
-295
------------
============
![Page 164: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/164.jpg)
• Here is the graphic depiction of our subtracting each data point from the mean (417):
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
200
Final Exam
Mid
term
Exa
m
200-417
= -217
417
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
417
417
417
417
417
417
417
417
417
417
417
417
Difference
-117
-97
-47
63
143
223
303
183
-17
-117
-217
-295
------------
============
![Page 165: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/165.jpg)
• Here is the graphic depiction of our subtracting each data point from the mean (417):
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
200-417
= +303
417
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
417
417
417
417
417
417
417
417
417
417
417
417
Difference
-117
-97
-47
63
143
223
303
183
-17
-117
-217
-295
------------
============
![Page 166: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/166.jpg)
• Here is the graphic depiction of our subtracting each data point from the mean (417):
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
200-417
= +303
417
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
417
417
417
417
417
417
417
417
417
417
417
417
Difference
-117
-97
-47
63
143
223
303
183
-17
-117
-217
-295
------------
============
![Page 167: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/167.jpg)
• Now we have the difference between the actual values for Y (ice cream sales) and the mean of the values for Y (417)
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
417
417
417
417
417
417
417
417
417
417
417
417
Difference
-117
-97
-47
63
143
223
303
183
-17
-117
-217
-295
------------
============
![Page 168: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/168.jpg)
• Now we have the difference between the actual values for Y (ice cream sales) and the mean of the values for Y (417)
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
417
417
417
417
417
417
417
417
417
417
417
417
Difference
-117
-97
-47
63
143
223
303
183
-17
-117
-217
-295
------------
============
![Page 169: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/169.jpg)
• As we showed previously we have to square this value because if we don’t when we sum the differences they will come to zero.
![Page 170: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/170.jpg)
• As we showed previously we have to square this value because if we don’t when we sum the differences they will come to zero.
Difference
-117
-97
-47
63
143
223
303
183
-17
-117
-217
-295
Squared
13689
9409
2209
3969
20449
49729
91809
33489
289
13689
47089
87025
SUM
= 0
SUM
= 372,844
![Page 171: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/171.jpg)
• As we showed previously we have to square this value because if we don’t when we sum the differences they will come to zero.
Difference
-117
-97
-47
63
143
223
303
183
-17
-117
-217
-295
Squared
13689
9409
2209
3969
20449
49729
91809
33489
289
13689
47089
87025
SUM
= 0
SUM
= 372,844
• We are doing all this once again to show a visual depiction of what the total sums of squares are:
![Page 172: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/172.jpg)
• As we showed previously we have to square this value because if we don’t when we sum the differences they will come to zero.
Difference
-117
-97
-47
63
143
223
303
183
-17
-117
-217
-295
Squared
13689
9409
2209
3969
20449
49729
91809
33489
289
13689
47089
87025
SUM
= 0
SUM
= 372,844
• We are doing all this once again to show a visual depiction of what the total sums of squares are:
Sum of Squares
df Mean Square F-ratio Significance
Total 372,844
![Page 173: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/173.jpg)
• Now that we’ve seen a visual depiction of how we calculated total sums of squares we compare the sums of squares that are associated with error (residual) and those associated with regression.
![Page 174: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/174.jpg)
• Now that we’ve seen a visual depiction of how we calculated total sums of squares we compare the sums of squares that are associated with error (residual) and those associated with regression.
Sum of Squares
df Mean Square F-ratio Significance
Regression Residual Total 372,844
![Page 175: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/175.jpg)
• Now that we’ve seen a visual depiction of how we calculated total sums of squares we compare the sums of squares that are associated with error (residual) and those associated with regression.
• Let’s calculate the error or residual sums of squares now.
Sum of Squares
df Mean Square F-ratio Significance
Regression Residual Total 372,844
![Page 176: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/176.jpg)
• The error or residual sums of squares are computed by subtracting each actual Y value from each Y predicted value.
![Page 177: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/177.jpg)
• The error or residual sums of squares are computed by subtracting each actual Y value from each Y predicted value.• Here are the actual Y values
![Page 178: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/178.jpg)
• The error or residual sums of squares are computed by subtracting each actual Y value from each Y predicted value.• Here are the actual Y values
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
These are the actual Y values or average ice cream sales
aver
age
ice
crea
m s
ales
![Page 179: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/179.jpg)
• The error or residual sums of squares are computed by subtracting each actual Y value from each Y predicted value.• Here are the actual Y values
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
These are the actual Y values or average ice cream sales
aver
age
ice
crea
m s
ales
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
![Page 180: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/180.jpg)
• Here are the predicted values using the linear regression formula:
![Page 181: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/181.jpg)
• Here are the predicted values using the linear regression formula:
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
These are the ac-tual Y values or average ice cream sales
aver
age
ice
crea
m s
ales
300
320
370
480
560
640
720
600
400
300
200
122
(y) Actual Ave Monthly Ice Cream Sales
= -50.93+7.21(300) == -50.93+7.21(320) =
= -50.93+7.21(480) =
= -50.93+7.21(370) =
= -50.93+7.21(560) =
= -50.93+7.21(640) =
= -50.93+7.21(720) =
= -50.93+7.21(600) =
= -50.93+7.21(400) == -50.93+7.21(300) =
= -50.93+7.21(200) == -50.93+7.21(122) =
Predicted Ave Monthly Ice Cream
Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
![Page 182: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/182.jpg)
• Here are the predicted values using the linear regression formula:
300
320
370
480
560
640
720
600
400
300
200
122
(y) Actual Ave Monthly Ice Cream Sales
= -50.93+7.21(300) == -50.93+7.21(320) =
= -50.93+7.21(480) =
= -50.93+7.21(370) =
= -50.93+7.21(560) =
= -50.93+7.21(640) =
= -50.93+7.21(720) =
= -50.93+7.21(600) =
= -50.93+7.21(400) == -50.93+7.21(300) =
= -50.93+7.21(200) == -50.93+7.21(122) =
Predicted Ave Monthly Ice Cream
Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
aver
age
ice
crea
m s
ales
![Page 183: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/183.jpg)
• Here are the predicted values using the linear regression formula:
300
320
370
480
560
640
720
600
400
300
200
122
(y) Actual Ave Monthly Ice Cream Sales
= -50.93+7.21(300) == -50.93+7.21(320) =
= -50.93+7.21(480) =
= -50.93+7.21(370) =
= -50.93+7.21(560) =
= -50.93+7.21(640) =
= -50.93+7.21(720) =
= -50.93+7.21(600) =
= -50.93+7.21(400) == -50.93+7.21(300) =
= -50.93+7.21(200) == -50.93+7.21(122) =
Predicted Ave Monthly Ice Cream
Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
aver
age
ice
crea
m s
ales
![Page 184: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/184.jpg)
• From these points and the linear regression formula a line can be drawn
![Page 185: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/185.jpg)
• From these points and the linear regression formula a line can be drawn
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
aver
age
ice
crea
m s
ales
![Page 186: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/186.jpg)
• From these points and the linear regression formula a line can be drawn
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
aver
age
ice
crea
m s
ales
![Page 187: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/187.jpg)
• The difference between each actual value (orange) and the predicted value (green line) is what is called error or residual. The closer these two values are to each other the smaller the error. The farther these two values are from each other the larger the error and the weaker the predictive power of the regression line.
![Page 188: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/188.jpg)
• The difference between each actual value (orange) and the predicted value (green line) is what is called error or residual. The closer these two values are to each other the smaller the error. The farther these two values are from each other the larger the error and the weaker the predictive power of the regression line.
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
aver
age
ice
crea
m s
ales
Difference
Difference
![Page 189: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/189.jpg)
• Let’s subtract the orange actual values and the green line predicted values:
![Page 190: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/190.jpg)
• Let’s subtract the orange actual values and the green line predicted values:
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
aver
age
ice
crea
m s
ales
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
Difference
62.53
10.43
-11.67
26.23
34.13
42.03
49.93
2.03
-125.87
-81.67
-37.47
28.73
------------
============
+28.73122
93
![Page 191: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/191.jpg)
• Let’s subtract the orange actual values and the green line predicted values:
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
aver
age
ice
crea
m s
ales
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
Difference
62.53
10.43
-11.67
26.23
34.13
42.03
49.93
2.03
-125.87
-81.67
-37.47
28.73
------------
============
-125.87
525
400
![Page 192: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/192.jpg)
• Let’s subtract the orange actual values and the green line predicted values:
• And so on…
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
aver
age
ice
crea
m s
ales
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
Difference
62.53
10.43
-11.67
26.23
34.13
42.03
49.93
2.03
-125.87
-81.67
-37.47
28.73
------------
============
-125.87
525
400
![Page 193: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/193.jpg)
• We then square those difference (deviations)
![Page 194: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/194.jpg)
• We then square those difference (deviations)
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
Difference
62.53
10.43
-11.67
26.23
34.13
42.03
49.93
2.03
-125.87
-81.67
-37.47
28.73
------------
============
Squared
3910.00
108.78
136.19
688.01
1164.86
1766.52
2493.00
4.12
15843.26
6669.99
1404.00
825.41
![Page 195: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/195.jpg)
• We then square those difference (deviations) and sum them up
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
Difference
62.53
10.43
-11.67
26.23
34.13
42.03
49.93
2.03
-125.87
-81.67
-37.47
28.73
------------
============
Squared
3910.00
108.78
136.19
688.01
1164.86
1766.52
2493.00
4.12
15843.26
6669.99
1404.00
825.41
Sum up
= 35,014
![Page 196: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/196.jpg)
Sum of Squares
df Mean Square F-ratio Significance
Regression Residual 35,014 Total 372,844
![Page 197: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/197.jpg)
• We will now calculate the regression sums of squares.
![Page 198: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/198.jpg)
• We will now calculate the regression sums of squares.
• Our hope is that this value will be much bigger than the residual (35,014).
Sum of Squares
df Mean Square F-ratio Significance
Regression Residual 35,014 Total 372,844
![Page 199: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/199.jpg)
• The regression sums of squares is calculated by subtracting the predicted values from the mean.
![Page 200: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/200.jpg)
• The regression sums of squares is calculated by subtracting the predicted values from the mean.• Let’s see what this looks like visually. The green line is the
predicted values for Y or the regression line.
![Page 201: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/201.jpg)
• The regression sums of squares is calculated by subtracting the predicted values from the mean.• Let’s see what this looks like visually. The green line is the
predicted values for Y or the regression line.
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
aver
age
ice
crea
m s
ales
![Page 202: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/202.jpg)
• The regression sums of squares is calculated by subtracting the predicted values from the mean.• Let’s see what this looks like visually. The green line is the
predicted values for Y or the regression line.
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
aver
age
ice
crea
m s
ales
![Page 203: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/203.jpg)
• The regression sums of squares is calculated by subtracting the predicted values from the mean.• Let’s see what this looks like visually. The green line is the
predicted values for Y or the regression line.
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
aver
age
ice
crea
m s
ales
The blue line is the mean (417)
which is the best predictor absent
anything else.
![Page 204: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/204.jpg)
• You can probably already tell that it will be bigger because a simple way to calculate it is to subtract the residual (35,014) from the total (372,844).
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
aver
age
ice
crea
m s
ales
![Page 205: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/205.jpg)
• You can probably already tell that it will be bigger because a simple way to calculate it is to subtract the residual (35,014) from the total (372,844).• However, we will calculate it the long way so you can see what
is happening.
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
aver
age
ice
crea
m s
ales
![Page 206: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/206.jpg)
• We subtract each predicted value from the mean of the actual Y values
![Page 207: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/207.jpg)
(y) Actual Ave Monthly Ice Cream Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
Mean Monthly Ice Cream Sales
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
Difference
-180.2
-108.1
-36.0
36.1
108.2
180.3
252.4
180.3
108.2
-36.0
-180.2
-324.4
------------
============
• We subtract each predicted value from the mean of the actual Y values
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
aver
age
ice
crea
m s
ales
![Page 208: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/208.jpg)
• We subtract each predicted value from the mean of the actual Y values
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
aver
age
ice
crea
m s
ales
(y) Actual Ave Monthly Ice Cream Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
Mean Monthly Ice Cream Sales
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
Difference
-180.2
-108.1
-36.0
36.1
108.2
180.3
252.4
180.3
108.2
-36.0
-180.2
-324.4
------------
============
93- 417- 324
![Page 209: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/209.jpg)
• We subtract each predicted value from the mean of the actual Y values
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
aver
age
ice
crea
m s
ales
(y) Actual Ave Monthly Ice Cream Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
Mean Monthly Ice Cream Sales
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
Difference
-180.2
-108.1
-36.0
36.1
108.2
180.3
252.4
180.3
108.2
-36.0
-180.2
-324.4
------------
============
670- 417+252
![Page 210: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/210.jpg)
• Then we square the differences (or deviations)
![Page 211: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/211.jpg)
• Then we square the differences (or deviations)
(y) Actual Ave Monthly Ice Cream Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
Mean Monthly Ice Cream Sales
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
Difference
-180.2
-108.1
-36.0
36.1
108.2
180.3
252.4
180.3
108.2
-36.0
-180.2
-324.4
------------
============
Squared
32470.8
11684.9
1295.76
1303.45
11708
32509.3
63707.4
32509.3
11708
1295.76
32470.8
105233
![Page 212: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/212.jpg)
• Then we square the differences (or deviations) and sum them up
(y) Actual Ave Monthly Ice Cream Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
Mean Monthly Ice Cream Sales
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
Difference
-180.2
-108.1
-36.0
36.1
108.2
180.3
252.4
180.3
108.2
-36.0
-180.2
-324.4
------------
============
Squared
32470.8
11684.9
1295.76
1303.45
11708
32509.3
63707.4
32509.3
11708
1295.76
32470.8
105233
Sum up
= 337,830
![Page 213: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/213.jpg)
• Then we square the differences (or deviations) and sum them up
Sum of Squares
df Mean Square F-ratio Significance
Regression 337,830 Residual 35,014 Total 372,844
![Page 214: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/214.jpg)
• Now we have all of the information to test for significance
![Page 215: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/215.jpg)
• Now we have all of the information to test for significance
Sum of Squares
df Mean Square F-ratio Significance
Regression 337,830 Residual 35,014 Total 372,844
![Page 216: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/216.jpg)
• The degrees of freedom (df) for the regression are the number of parameters that are being estimated which in this case is the Y intercept and the slope in this equation minus
![Page 217: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/217.jpg)
• The degrees of freedom (df) for the regression are the number of parameters that are being estimated which in this case is the Y intercept and the slope in this equation minus • 2 parameters -1 = 1
Sum of Squares
df Mean Square F-ratio Significance
Regression 337,830 1
Residual 35,014 Total 372,844
![Page 218: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/218.jpg)
• The degrees of freedom for residual is the number of cases (12) minus the number of parameters (2)
![Page 219: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/219.jpg)
• The degrees of freedom for residual is the number of cases (12) minus the number of parameters (2)• 12 months – 2 parameters (slope / y intercept) = 10
![Page 220: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/220.jpg)
• The degrees of freedom for residual is the number of cases (12) minus the number of parameters (2)• 12 months – 2 parameters (slope / y intercept) = 10
Sum of Squares
df Mean Square F-ratio Significance
Regression 337,830 1
Residual 35,014 10 Total 372,844
![Page 221: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/221.jpg)
• We now have the information we need to calculate the Mean Square values. They are calculated by dividing the sums of squares by the degrees of freedom.
![Page 222: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/222.jpg)
• We now have the information we need to calculate the Mean Square values. They are calculated by dividing the sums of squares by the degrees of freedom.
Sum of Squares
df Mean Square F-ratio Significance
Regression 337,830 1 =337,830
Residual 35,014 10 =3,501
Total 372,844
![Page 223: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/223.jpg)
• The F-ratio is computed by dividing the Regression Mean Square by the Residual Mean Square
![Page 224: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/224.jpg)
• The F-ratio is computed by dividing the Regression Mean Square by the Residual Mean Square
• 337,830 / 3,501 = 96.5
![Page 225: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/225.jpg)
• The F-ratio is computed by dividing the Regression Mean Square by the Residual Mean Square
• 337,830 / 3,501 = 96.5
Sum of Squares
df Mean Square F-ratio Significance
Regression 337,830 1 337,830 96.5
Residual 35,014 10 3,501
Total 372,844
![Page 226: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/226.jpg)
• With this information we can turn to the F-distribution table to determine the significance value.
![Page 227: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/227.jpg)
• With this information we can turn to the F-distribution table to determine the significance value.
Sum of Squares
df Mean Square F-ratio Significance
Regression 337,830 1 337,830 96.5
Residual 35,014 10 3,501
Total 372,844
![Page 228: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/228.jpg)
Sum of Squares
df Mean Square F-ratio Significance
Regression 337,830 1 337,830 96.5
Residual 35,014 10 3,501
Total 372,844
![Page 229: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/229.jpg)
• The regression degrees of freedom (1) is represented by the columns below:
Sum of Squares
df Mean Square F-ratio Significance
Regression 337,830 1 337,830 96.5
Residual 35,014 10 3,501
Total 372,844
![Page 230: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/230.jpg)
• The regression degrees of freedom (1) is represented by the columns below:
Sum of Squares
df Mean Square F-ratio Significance
Regression 337,830 1 337,830 96.5
Residual 35,014 10 3,501
Total 372,844
![Page 231: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/231.jpg)
Sum of Squares
df Mean Square F-ratio Significance
Regression 337,830 1 337,830 96.5
Residual 35,014 10 3,501
Total 372,844
![Page 232: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/232.jpg)
• The residual degrees of freedom (10) is represented by the rows below:
Sum of Squares
df Mean Square F-ratio Significance
Regression 337,830 1 337,830 96.5
Residual 35,014 10 3,501
Total 372,844
![Page 233: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/233.jpg)
• The residual degrees of freedom (10) is represented by the rows below:
Sum of Squares
df Mean Square F-ratio Significance
Regression 337,830 1 337,830 96.5
Residual 35,014 10 3,501
Total 372,844
![Page 234: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/234.jpg)
Sum of Squares
df Mean Square F-ratio Significance
Regression 337,830 1 337,830 96.5
Residual 35,014 10 3,501
Total 372,844
![Page 235: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/235.jpg)
• Put them together and we have found the critical F value at the .05 alpha level to be 4.96.
Sum of Squares
df Mean Square F-ratio Significance
Regression 337,830 1 337,830 96.5
Residual 35,014 10 3,501
Total 372,844
![Page 236: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/236.jpg)
• Put them together and we have found the critical F value at the .05 alpha level to be 4.96.
Sum of Squares
df Mean Square F-ratio Significance
Regression 337,830 1 337,830 96.5
Residual 35,014 10 3,501
Total 372,844
![Page 237: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/237.jpg)
• Because the F-ratio (96.5) exceeds the F-critical (4.96) we will reject the null hypothesis and indicate that temperature is a statistically significant predictor of ice cream sales
![Page 238: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/238.jpg)
In Summary
![Page 239: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/239.jpg)
In Summary
• The whole point of this demonstration was to
![Page 240: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/240.jpg)
In Summary
• The whole point of this demonstration was to (1) explain that linear regression is used to predict the value of one variable (ice cream sales) based on another variable (temperature)
![Page 241: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/241.jpg)
In Summary
• The whole point of this demonstration was to (1) explain that linear regression is used to predict the value of one variable (ice cream sales) based on another variable (temperature)(2) show that the total variance in Y can be partitioned into regression (prediction power) and residual (error)
![Page 242: Single linear regression](https://reader038.vdocument.in/reader038/viewer/2022102902/5584791ed8b42a6b4d8b51fb/html5/thumbnails/242.jpg)
In Summary
• The whole point of this demonstration was to (1) explain that linear regression is used to predict the value of one variable (ice cream sales) based on another variable (temperature)(2) show that the total variance in Y can be partitioned into regression (prediction power) and residual (error) (3) show how this can be used to test whether the prediction is better than by chance.