1. Calculating the Least Squares Regression Line
Lecture 49
Secs
Wed, Dec 7, 2005

2. The Least Squares Regression Line
The equation of the regression line is
y^ = a + bx.
Thus, we need to find the coefficients a and b.
The formulas are or

3. Example
Consider again the data set x y 2 3 5 9 6 12 16

4. Method 1 Compute the means and deviations for x and y. x y x –x y –y2 3 -3 -6 5 -2 -4 9 6 12 1 16 4 7
x = 5
y = 9

5. Method 1 Compute the squared deviations, etc. x y 2 3 -3 -6 9 36 18 5
x –x
y –y
(x –x)2
(y –y)2
(x –x)(y –y) 2 3 -3 -6 9 36 18 5 -2 -4 4 16 8 6 12 1 7 49 28

6. Method 1 Find the sums of the last three columns. x y 2 3 -3 -6 9 36
x –x
y –y
(x –x)2
(y –y)2
(x –x)(y –y) 2 3 -3 -6 9 36 18 5 -2 -4 4 16 8 6 12 1 7 49 28 30
110 57

7. Method 1
Compute b:
Then compute a:

8. Method 2
Consider again the data x y 2 3 5 9 6 12 16

9. Method 2 Compute x2, y2, and xy for each row. x y x2 y2 xy 2 3 4 9 6 525 15 81 45 12 36
144 72 16
256

10. Method 2 Then find the sums of x, y, x2, y2, and xy. x y x2 y2 xy 2 34 9 6 5 25 15 81 45 12 36
144 72 16
256 25 45
155
515
282

11. Method 2 Then find the sums of x, y, x2, y2, and xy. x y x2 y2 xy 2 34 9 6 5 25 15 81 45 12 36
144 72 16
256
x = 25
y = 45
x2 = 155
y2 = 515
xy = 282 25 45
155
515
282

12. Method 2
Compute b:
Then compute a:

13. Example The second method is usually easier.
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 the formulas for a and b.
For our example, 2-Var Stats L1, L2 reports that
n = 5
x = 25
x2 = 155
y = 45
y2 = 515
xy = 282

15. TI-83 – Regression Line Or we can use the LinReg function.
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.

16. 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).

17. Let’s Do It! Let’s Do It! 13.3, p. 819 – Oil Change Data.
Use the TI-83.
Enter the data into lists L1 and L2.
Draw the scatterplot in order to decide whether a linear model is reasonable.
Compute the regression line.