1. BASIC REGRESSION CONCEPTS

2. Basic Idea Behind Regression
To to determine how a particular variable (called the dependent variable -- y) is influenced by one or more other variables (called independent variables -- x1, x2, etc.) x1 x2 x3 y

3. 5-step Regression Approach
Hypothesize a form of the model
Hypothesize whether a linear relation, quadratic relation, etc. exists between y and the x’s
Determine the best estimates for the values of the parameters
Make and check any necessary assumptions
Evaluate the model
Determine how good the model is
If the model is good -- use it for prediction and estimation

4. Confidence Interval Review
Suppose sales at Dollar Only Stores for 10 randomly selected weeks were:
Week Sales
1 101,000
,000
3 110,000
4 120,000
,000
,000
,000
,000
,000
,000

5. Confidence Interval for Average Weekly Sales
Assuming that weekly sales are normally distributed, a 95% confidence interval for average weekly sales is:

6. Using Excel
E3-F16
E3+F16

7. Regression Concepts
But couldn’t the sales (y) have been affected by one or more factors?
Advertising dollars (x1)
Average number of salesmen (x2)
Hours of operation (x3)
The weather (good or not good) -- (x4)
In this case we may want to “regress” y on one or more of these variables

8. The Basic Linear Regression Relation
We might hypothesize that sales are linearly dependent on all four of the previous variables, i.e.:
y = 0 + 1x1 + 2x2 + 3x3 + 4x4 + 
|<=======Regression =======>| |Error|
The ’s are (unknown) constants
We shall estimate them by b0, b1, b2, b3 and b4
 is a random variable for the variability (error) when x1, x2, x3, and x4 take on a specific set of values
 has a distribution, a mean, and a standard deviation

9. Simple Linear Regression
Simple linear regression is when we regress
y (Sales) on only one variable x (Ad $)
y = 0 + 1x + 
Here, 0 = the true value of the y-intercept
1 = the true slope of the line
 = a random variable of the “error”

10. INPUT DATA
The Dollar Only Stores advertising for the corresponding 10 sample weeks is:
Week(i) Ad $ (xi) Sales (yi)
,000
,000
,000
,000
,000
,000
,000
,000
,000
,000

11. Step 1 -- Hypothesizing the form of the model
If we are regressing on only one variable -- use a scatterplot to determine an appropriate model
Does it look like the data is relatively linear?
y = 0 + 1x + 
Does it look curved?
Perhaps y = 0 + 1x + 2x2 + 
etc.
LET’S SEE!

12. Scatterplot

13. Step 1
It looks like a straight line fits through the points fairly well.
Thus, we hypothesize:
y = 0 + 1x + 
We now must get the best estimates for 0 and 1 -- This is step 2!

14. Review
Regression seeks to explain how a dependent variable (y) is affected by independent variables (x1, x2, x3, etc.)
Regression is a multi-step procedure.
The first step is to hypothesize a form of the model.
If there is only one variable, plot y vs. x to assist in forming the hypothesis.