1. Regression & Prediction
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Regression & Prediction

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Linear Regression
Finding the best fitting straight line for a set of data
This line is represented by the equation
Y = bX + a, where a & b are fixed constants

3. Least Squares Regression
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Least Squares Regression
Most commonly used regression line
Makes the sum of the squared errors as small as possible
Regression Line Equation
Y=bX+a
Y(hat) is the predicted value of Y given a certain X
b is the slope
a is the y-intercept

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Slope (b)
Indicates by how much Y will change when X is increased by one point

5. Relationship of “b” to “r”
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Relationship of “b” to “r”
When “r” is positive, “b” is positive
When “r” is zero, “b” is zero
When “r” is negative, “b” is negative

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Y-Intercept (a)
a = MY - bMX
Value of Y when X is equal to 0

7. How Good is Our Prediction - Standard Error of Estimate
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How Good is Our Prediction - Standard Error of Estimate
Provides a measure of how accurately the regression equation predicts the Y values
Gives a measure of the standard distance between a regression line and the actual data points
Analogous to standard deviation.

8. Standard Error of Estimate - Formula
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Standard Error of Estimate - Formula

9. Another Formula for Standard Error of Estimate
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Another Formula for Standard Error of Estimate
SSerror=(1-r2)SSY
By using this formula for SSerror the standard error of estimate can also be computed as

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Example
Exam 1 (X)
Final Grades (Y) 62 74 73 93 88 68 82 79 85 91 77 72 94 96 65 61 92 98 95
A professor claims that the scores on the 1st exam provide an excellent indication of how students will perform throughout the term

11. Exam 1 (x) Final Grades (Y) X2 Y2 XY 62 74 3844 5476 4588 73 93 5329
8649
6789 88 68
7744
4624
5984 82 79
6724
6241
6478 85 91
7225
8281
7735 77 72
5929
5184
5544 94 96
8836
9216
9024 65 61
4225
3721
3965 92
8464
8372
6068
7905 98 95
9604
9025
9310
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Calculate SP

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Calculate SSx
And the SSx

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Calculate SSY
And the SSy

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Calculate r

16. Calculate Regression Equation
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Calculate Regression Equation
Y’ = bX+a
a = MY - bMX

17. Standard Error of Estimate
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Exam 1 (x)
Final Grades (Y)
Predicted Final
(Y-Y’)
(Y-Y')2 62 74
70.35
3.65 73 93
77.61
-15.39 88 68
87.51
-19.51 82 79
83.55
-4.55 85 91
85.53
5.47 77 72
80.25
-8.25 94 96
91.47
4.53 65 61
72.33
-11.33 92
89.49
2.51
6.3001
78.27
3.73
7.47 98 95
94.11
0.89
0.7921
We know now that our predictions for y based on x will not be perfect, however we can calculate approximately how far off our predictions will be.
That is, we need is is called the standard error of the estimate.
Our regression equation is: Y’ = 0.66X+29.43
We can do this by finding the predicted value for each X value and then subtracting Y-hat from Y and squaring the deviation, and then dividing it by its degree of freedom to obtain a measure of variance and taking the square root

18. Standard Error of Estimate

19. Multiple Regression: Some Additional Points
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Multiple Regression: Some Additional Points
If b1>b2, does that mean that X1 is a better predictor than X2?