1. BA 275 Quantitative Business Methods
Agenda
Simple Linear Regression
Inference for Regression
Inference for Prediction

2. Regression Analysis
A technique to examine the relationship between an outcome variable (dependent variable, Y) and a group of explanatory variables (independent variables, X1, X2, … Xk).
The model allows us to understand (quantify) the effect of each X on Y.
It also allows us to predict Y based on X1, X2, …. Xk.

3. Types of Relationship Linear Relationship Nonlinear Relationship
Simple Linear Relationship
Y = b0 + b1 X + e
Multiple Linear Relationship
Y = b0 + b1 X1 + b2 X2 + … + bk Xk + e
Nonlinear Relationship
Y = a0 exp(b1X+e)
Y = b0 + b1 X1 + b2 X12 + e
… etc.
Will focus only on linear relationship.

4. Simple Linear Regression Model
population
True effect of X on Y
Estimated effect of X on Y
sample
Key questions:
1. Does X have any effect on Y?
2. If yes, how large is the effect?
3. Given X, what is the estimated Y?

5. Least Squares Method Least squares line:
It is a statistical procedure for finding the “best-fitting” straight line.
It minimizes the sum of squares of the deviations of the observed values of Y from those predicted
Sum of Squares is minimized.
Bad fit.

6. Initial Analysis
Summary statistics + Plots (e.g., histograms + scatter plots) + Correlations
Things to look for
Features of Data (e.g., data range, outliers)
do not want to extrapolate outside data range because the relationship is unknown (or un-established).
Summary statistics and graphs.
Is the assumption of linearity appropriate?

7. Correlation
r (rho): Population correlation (its value most likely is unknown.)
r: Sample correlation (its value can be calculated from the sample.)
Correlation is a measure of the strength of linear relationship.
Correlation falls between –1 and 1.
No linear relationship if correlation is close to 0.
r = – –1 < r < r = < r < r = 1
r = – –1 < r < r = < r < r = 1

8. Correlation (r vs. r) Sample size P-value for H0: r = 0 Ha: r ≠ 0

9. Fitted Model: Least Squares Lineb0 b1
Least squares line: estimated_Price = – Area.

10. Hypothesis Testing Key Q1: Does X have any effect on Y?b0
H0: b1 = 0
Ha: b1 ≠ 0 b1
SEb1
SEb0
Degrees of freedom = n – p – 1
p = # of independent variables used.

11. Interval Estimation Key Q2: If so, how large is the effect?b0 b1
SEb1
SEb0
Degrees of freedom = n – p – 1
p = # of independent variables used.

12. Prediction and Confidence Intervals Key Q3: Given X, what is the estimated Y?
What is your estimated price of that 2000-sf house on the 9th street?
Quick answer: estimated price = (2) =
What is the average price of a house that occupies 2000 sf?
What is the difference?

13. Prediction and Confidence Intervals

14. Prediction and Confidence Intervals
Prediction interval
Confidence interval

15. Model Comparison: A Good Fit?
SS = Sum of Squares = ???