1. Simple Linear Regression & Correlation
(X1,Y1) (X2,Y2) (X3,Y3) (X4,Y4) … (Xn,Yn)
Relationship Between Two Variables:
Linear Relationship Curvilinear Relationship No Relationship + + + + + + + + + + + + + + + +
1) Measure the Degree of Association Between the 2 Variables
2) Forecast Future Values
3) Measure the Error

2. Simple Linear Regression

3. Method of Least Squares
Want to Minimize this
This leads to 2 Equations:

4. Example 1: Test scores 1st 2nd
X Y
80 90
60 70
40 40
30 40
40 60

6. Example 2: Ads vs Sales Ads Sales
X Y
0 1
1 2
2 3
3 5
4 8
5 11
6 12

8. Partition the Sum of Squares:
Total Variation of Y
Total = Unexplained Explained
Variation Variation Variation
TSS = SSE SSR

9. Regression ANOVA Table:
Source df SS MS
Regression 1 SSR = b1SSxy MSR = SSR/1
Error n-2 SSE = SSy - b1SSxy MSE = SSE/n-2
Total n-1 TSS = SSy
Coefficient of Determination:
r2 = SSR/TSS % of Variation of Y Explained by X
Model Hypothesis Test:
H0: Model is Not Significant
HA: Model is Significant
R: F > Fα(1,n-2) F = MSR/MSE

10. Ex 1:
Ex 2:

11. MSE = Estimated Error Variance Se2
Parameter Estimator (Standard Error)2
β0 b0
β1 b1
Interval Estimate: b0 - e ≤ β0 ≤ b0 + e ; e = t•Sb0
Ex 1:

13. Interval Estimate: b1 - e ≤ β1 ≤ b1 + e ; e = t•Sb1
Ex 2:
Hypothesis Test: Ex 2:
H0: β1 = 0
HA: β1 ≠ 0
R: t > tα/2,df=n-2
t < -tα/2,df=n-2

14. Parameter Estimator (Std Error)2Y
E(Y|X)

15. Confidence Interval for the Mean Value of Y for a Given X
Ex 2: Xg = 5

16. Interval Estimate for a Single Value of Y for a Given X
Ex 2: Xg = 5
Prediction Interval

17. Correlation Coefficient: -1 ≤ r ≤ 1

18. Sample Correlation
Coefficient:
Ex1:
Ex 2:
Hypothesis Test:
H0: ρ = 0
HA: ρ ≠ 0
R: t > tα/2,df=n-2
t < -tα/2,df=n-2
Ex 2:

19. Ex 3: Units Cost
X2 X XY Y Y2
1 8
2 8
3 10
4 14
5 16
6 16
7 18
8 22
SSx =
SSxy =
SSy =

20. Prediction Equation:
ANOVA Table for the Model:
Test the Model:

21. Test for β1 = 0:
95% CI for β0:
Test for ρ = 0:

22. 95% CI for Mean Value of Y when X = 7:
95% PI for Single Value of Y when X = 7:

23. Human Resource Selection Ho: Other Performer – Do Not Hire
HA: Superior Performer – Do Hire
R: Test Score > Xc
Superior
False Negative
False positive
Other
Do Not Hire
Do Hire