1. Calculating the Least Squares Regression Line
Lecture 45
Sec
Fri, Nov 30, 2007

2. The Least Squares Regression Line
The equation of the regression line is of the form
y^ = a + bx.
We need to find the coefficients a and b from the data.

3. The Least Squares Regression Line
The formula for b is
The formula for a is or

4. Formula 1 Consider again the data set x y 1 8 3 12 4 9 5 14 16 20 1117 15 24

5. Formula 1 Compute the x and y deviations. x y x –x y –y 1 8 -6 -7 312 -4 -3 4 9 5 14 -2 -1 16 20 2 11 17 15 24

6. Formula 1 Compute the squared deviations. x y x –x y –y (x –x)28 -6 -7 36 49 42 3 12 -4 -3 16 9 4 18 5 14 -2 -1 2 20 25 10 11 17 15 24 64 81 72

7. Formula 1 Find the sums. x y x –x y –y (x –x)2 (y –y)2 1 8 -6 -736 49 42 3 12 -4 -3 16 9 4 18 5 14 -2 -1 2 20 25 10 11 17 15 24 64 81 72
150
206
165

8. Formula 1
Compute b:
Then compute a:
The equation is

9. Formula 2 Consider yet again the data set x y 1 8 3 12 4 9 5 14 16 2011 17 15 24

10. Formula 2 Square x and y and find xy. x y x2 y2 xy 1 8 64 3 12 9 14436 4 16 81 5 14 25
196 70
256
128 20
400
180 11 17
121
289
187 15 24
225
576
360

11. Formula 2 Add up the columns. x y x2 y2 xy 1 8 64 3 12 9 144 36 4 1681 5 14 25
196 70
256
128 20
400
180 11 17
121
289
187 15 24
225
576
360 56
120
542
2006
1005

12. Method 2
Compute b:
Then compute a as before:
The equation is

13. Example
The second method is usually easier (really) if you are doing it by hand.
By either method, we get the equation
y^ = x.

14. TI-83 – Regression Line
On the TI-83, we could use 2-Var Stats to get the basic summations. Then use Formula 2 for b.
Try it: Enter 2-Var Stats L1, L2.

15. TI-83 – Regression Line 2-Var Stats L1, L2 reports that
x = 56
x2 = 542
y = 120
y2 = 2006
xy = 1005
Then use the formulas.

16. TI-83 – Regression Line Or we can use the LinReg function (#8).
Put the x values in L1 and the y values in L2.
Select STAT > CALC > LinReg(a+bx).
Press Enter. LinReg(a+bx) appears in the display.
Enter L1, L2.
Press Enter.

17. TI-83 – Regression Line The following appear in the display.
The title LinReg.
The equation y = a + bx.
The value of a.
The value of b.
The value of r2 (to be discussed later).
The value of r (to be discussed later).

18. TI-83 – Regression Line
To graph the regression line along with the scatterplot,
Put the x values in L1 and the y values in L2.
Select STAT > CALC > LinReg(a+bx).
Press Enter. LinReg(a+bx) appears in the display.
Enter L1, L2, Y1
Press Enter.

19. TI-83 – Regression Line
To graph the regression line along with the scatterplot,
Put the x values in L1 and the y values in L2.
Select STAT > CALC > LinReg(a+bx).
Press Enter. LinReg(a+bx) appears in the display.
Enter L1, L2, Y1
Press Enter.
Add this

20. TI-83 – Regression Line Press Y= to see the equation.
Press ZOOM > ZoomStat to see the graph.

21. Free Lunch Participation vs. Graduation Rate
Find the equation of the regression line for the school-district data on the free-lunch participation rate vs. the graduation rate.
Let x be the free-lunch participation.
Let y be the graduation rate.

22. Free Lunch Participation vs. Graduation Rate
District
Free Lunch
Grad. Rate
Amelia
41.2
68.9
King and Queen
59.9
64.1
Caroline
40.2
62.9
King William
27.9
67.0
Charles City
45.8
67.7
Louisa
44.9
80.1
Chesterfield
22.5
80.5
New Kent
13.9
77.0
Colonial Hgts
25.7
73.0
Petersburg
61.6
54.6
Cumberland
55.3
63.9
Powhatan
12.2
89.3
Dinwiddie
45.2
71.4
Prince George
30.9
85.0
Goochland
23.3
76.3
Richmond
74.0
46.9
Hanover
13.7
90.1
Sussex
74.8
59.0
Henrico
30.2
81.1
West Point
19.1
82.0
Hopewell
63.1
63.4

23. Free Lunch Participation vs. Graduation Rate
The regression equation is
y^ = – 0.494x.

24. Scatter Plot 90 80 Graduation Rate 70 60 50 Free Lunch Rate 20 30 40

25. Scatter Plot with Regression Line90 80
Graduation Rate 70 60 50
Free Lunch
Rate 20 30 40 50 60 70 80

26. Predicting y
What graduation rate would we predict in a district if we knew that the free-lunch participation rate was 50%?

27. Scatter Plot with Regression Line90
Predicted
Point 80
Graduation Rate 70 60 50
Free Lunch
Rate 10 20 30 40 50 60 70 80

28. Scatter Plot with Regression Line90
Predicted
Point 80
Graduation Rate 70
66.3 60
Predicted
Graduation
Rate 50
Free Lunch
Rate 10 20 30 40 50 60 70 80

29. Variation in the Model
There is a very simple relationship between the variation in the observed y values and the variation in the predicted y values.

30. Observed y and Predicted y90
Observed y
Predicted y 80
Graduation Rate 70 60 50
Free Lunch
Rate 10 20 30 40 50 60 70 80

31. SST = Variation in the Observed y90 80
Graduation Rate 70 60 50
Free Lunch
Rate 10 20 30 40 50 60 70 80

32. SSR = Variation in the Predicted y90 80
Graduation Rate 70 60 50
Free Lunch
Rate 10 20 30 40 50 60 70 80

33. Variation in Observed y
The variation in the observed y is measured by SST (same as SSY).
For graduation rate data (L2),
SST =

34. Variation in Predicted y
The variation in the predicted y is measured by SSR.
For predicted graduation rate data, let
L3 = Y1(L1).
SSR =

35. SSE = Residual Sum of Squares
It turns out that
SST = SSE + SSR.
That is,

36. Sum Squared Error In the example, SST – SSR = 2598.18 – 1896.67=
If we compute the sum of the squared residuals directly, we get
SSE =

37. Explaining the Variability
In the equation
SST = SSE + SSR,
SSR is the amount of variability in y that is explained by the model.
SSE is the amount of variability in y that is not explained by the model.

38. Explaining the Variability
In the last example, how much variability in graduation rate is explained by the model (by free-lunch participation)?