A Line of Best Fit An Example A study was conducted on 20 drivers to determine the relationship between the drivers speed and the braking distance at the sight of an obstacle The following results were recorded from a sample of 20 drivers ID: 332293
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Slide1
Mayer Line
A Line of Best FitSlide2
An Example
A study was conducted on 20 drivers to determine the relationship between the drivers’ speed and the braking distance at the sight of an obstacle. The following results were recorded from a sample of 20 drivers
X
km/h
40
45
50
55
60
65
70
75
80
85
Y
m
2.80
3.30
3.75
4.2
5.05
6.10
6.60
7.20
7.80
8.60
X
90
95
100
105
110
115
120
125
130
135
Y
9.20
9.95
10.60
11.05
11.85
12.90
13.30
13.60
14.00
14.50Slide3
Create a Scatter Plot of DataSlide4
What do you notice?
There appears to be some kind of correlation between the speed and the braking distance.
What is the purpose of establishing a correlation in this particular example?Slide5
Regression Analysis
In statistics, regression analysis includes any techniques for modeling and analyzing variables, when the focus is on the relationship between a dependent variable and an independent variable Slide6
The Regression Line
A
Regression Line
is the straight line that BEST represents the set of data.
The EQUATION of the regression line enables you to predict with some accuracy the
y
value given a particular
x
value.Slide7
Procedure:
Put the ordered pairs into numerical order based on the
x
-values. Separate the scatter plot into two groups (G
1
and G
2
) containing the
SAME
number of points. If there are an odd number of data, let one group have an extra ordered pair.
For each group, calculate the mean of the
x
values and the mean of the
y
values. From this you will create two new points called the
mean points.
The Mayer Line is the line passing through these mean points so calculate the equation of the straight line passing through the mean points.
This line is a line of best fit for the scatter plot otherwise known as the regression lineSlide8
Divide the DataSlide9
Find Mean Points for LineSlide10
Find Equation of LineSlide11
Once you have the equation…
…make a prediction
A person driving 121 km/ hr, would have a braking distance of how much?
A person with a baking distance of 9 meters was driving how fast? Slide12
Caution
Using the regression line to predict a
y
value is only valid when the x
value being used is located in the variation interval of the variable
x
.
For this example, it means that a prediction can only
really
be made for a
y
value, if the
x
value lies between 40 and 135. You can predict the braking distance for a car traveling 89 km/h , but not for one traveling 19 km/h.
You may be asked to make predictions
beyond
the variation. You CAN do this but you should note that it will certainly decrease the validity of the analysis if you do so. Always use your common sense. Slide13
Homework
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Page 296 #3
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Page 297 #4 a)
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