1. Class 28
Get Ready….

2. Height and Weight Is CM or Inches the better predictor of KG?
Whichever has the lower standard error
Will also have a variety of better stats
NOT whichever has the bigger coefficient
A multiple regression lets you test
H0: all b’s = 0 (nothing in the model matters)
H0: b1=0 given all the other b’s
When using both CM and INCHES
We reject H0 b1=b2=0
We fail to reject H0 b1=0 given b2
We fail to reject H0 b2=0 given b1
You need either CM or INCHES but not both
Because they are highly correlated
Regressions ALWAYS go thru the sample averages

3. Things I expect you will know
How to interpret a regression using p-1 dummy variables
The p possible forecasts will equal the sample average Y for each of the p groups
The intercept is the average of the left-out group
The coefficients are differences in group averages.
The p-value/significance F will match that from ANOVA single factor

4. Things I expect you will know
How to interpret a residual (error)
It is Y - 𝑌
It is the distance each Y is from the line.
Positive means above the line.
They measure the difference between actual Y and expected Y (based on the X’s)
The most over-weight girl (for her height) is the girl with the largest positive residual.
Check the box to get residuals.

5. Things I expect you will know
How to interpret a coefficient in a multiple regression.
It measures the change in expected Y for a unit change in that X keeping all other Xs constant.
If I keep miles and stops constant and change from williams to spencer, expect 0.97 hours less.
If I change from Williams to Spencer, expect 0.33 hours more.
It is the easy way to answer some questions.
If the previous rating goes from 17.5 to 20, how will the expected ratings change? (by per point)

6. Things I expect you will know
How to use a regression model to calculate a point forecast.
Plug and chug.
I use SUMPRODUCT
You must know what Xs to plug in.
It is a package deal….you must know and plug in ALL the Xs.

7. Things I expect you will know
How to use a regression model to calculate a probability.
The question gives you the Y.
You Plug and chug to get the 𝑌 .
You calculate t = (Y - 𝑌 )/ standard error
Use t.dist.rt( t , dof)
Dof is n – total number of regression terms.
Requires the FOUR assumptions.

8. Things I expect you will know
If the coefficient of X1 changes when X2 is included in the model…..
You know X1 and X2 are correlated.
You can use the two regression results to tell whether X1 and X2 are positively or negatively correlated.
Ds was positively correlated with Miles
Fact was negatively correlated with Stars
Nobel was positively correlated with Yanks
Speed was positively correlated with Dcorporate
Exam 1 was negatively correlated with Exam 2.

9. Oh…Fact Movies had fewer Stars!
UNDERSTANDING
Coefficient
Regression Table
Constant
Fact
Coefficient
Regression Table
Constant
12.568
Fact
1.799
Stars
1.259
Oh…Fact Movies had fewer Stars!

10. Oh…Fact Movies had fewer Stars!
Secret Formula
Coefficient
Regression Table
Constant
Fact
Coefficient
Regression Table
Constant
12.568
Fact
1.799
Stars
1.259
Regress Y on X1
𝑐 = 𝑏 − 𝑏 𝑏 2
Regress Y on X1 and X2
Oh…Fact Movies had fewer Stars!
Regress Y on X1 and X2
Regress X2 on X1

11. Secret Formula 𝑐 = 1.40−1.80 1.26 𝑐 =−0.32 Regress Y on X1
Coefficient
Regression Table
Constant
Fact
Coefficient
Regression Table
Constant
12.568
Fact
1.799
Stars
1.259
Regress Y on X1
Regress Y on X1 and X2
𝑐 = 1.40−
Regress Y on X1 and X2
𝑐 =−0.32
Regress X2 on X1

12. Oh…Fact Movies had fewer Stars!
UNDDERSTANDING
Coefficient
Regression Table
Constant
Fact
Coefficient
Regression Table
Constant
12.568
Fact
1.799
Stars
1.259
Oh…Fact Movies had fewer Stars!

13. Fact Movies averaged 0.32 fewer Stars!
UNDERSTANDING
Secret Formula
Coefficient
Regression Table
Constant
Fact
Coefficient
Regression Table
Constant
12.568
Fact
1.799
Stars
1.259
Fact Movies averaged 0.32 fewer Stars!

14. Regression is the line through a cloud of points
Scatter-plot the cloud
It is up to YOU to interpret the results.
Don’t assume X causes Y
Y might be causing X
Both might be caused by Z
Don’t assume better fitting lines are better at forecasting
They usually are not…..too good a fit means too complicated a model…..means poorer performance.

15. Class 28 Assignment Variable School Graduation Rate
% of Classes Under 20
Student/Faculty Ratio
Alumni Giving Rate
Description
The name of the
University
Percentage of enrollees who graduate
Percentage of Classes offered with <= 20 students.
Number of students enrolled divided by total number of faculty
Percentage of living alumni who gave to the University in 2000
Mean
83.042
55.729
11.542
29.271
Median
83.5
59.5
10.5 29
Mode 92 65 13
Standard Deviation
8.607
13.194
4.851
13.441
Skewness
-0.282
-0.501
0.582
0.370
Minimum 66 3 7
Maximum 97 77 23 67
Count 48

16. Regress Giving Rate on Grad Rate Check if coeff is positive
1. Test the hypothesis that graduation rate and alumni giving rate are (linearly) independent. We expect universities with higher graduation rates to have higher mean giving rates. [15 points]
Regress Giving Rate on Grad Rate
Check if coeff is positive
Divide reported p-value (found in two places) by 2.
Reject if less than 0.05.
Coefficients
Standard Error
t Stat
P-value
Intercept
-68.76
12.58
-5.46
1.82E-06
Graduation Rate
1.18
0.15
7.83
5.24E-10

17. 2. If the graduation rate of school A is 5 percentage points higher than that of school B, how much higher do we expect school A’s giving rate to be? [10 points]
Using the above regression (graduation rate is all we know), the expected giving rate will be 1.18*5 = 5.9 percentage points higher for school A.

18. 3. If you learn that A and B above have identical student to faculty ratios, what is your revised answer to question 2? Be certain to explain why it went up (if it went up) or why it went down (if it went down) or why it stayed the same. Direct your response to a university administrator. [15 points]
Coefficients
Standard Error
t Stat
P-value
Intercept
Graduation Rate
Student/Faculty Ratio
IF we keep SFR constant, expected Giving Rate goes up 0.76 points per point of graduation rate.
If we don’t keep SFR constant, expected Giving Rates went up 1.18 points per point.
Schools with higher grad rates had LOWER SFR (that makes sense)
If we don’t hold SFR constant, increases in grad rate mean decreases in SFR and the combined effect of the two is 1.18.
So….if grad rate is higher (but SFR is not), expected 0.76 increase.
If grad rate is higher (and SFR is lower as in the data), expect 1.18 increase.

19. Don’t Use this variable.
4. Provide a point forecast of alumni giving rate for a university with graduation rate of 80, 65 percent of its classes with 20 or fewer students, and a student/faculty ratio of 20. [25 points]
Coefficients
Standard Error
t Stat
P-value
Intercept
0.2433
Graduation Rate
0.7482
0.1660
4.5082
0.0000
% of Classes Under 20
0.0290
0.1393
0.2084
0.8358
Student/Faculty Ratio
0.3867
0.0035
Don’t Use this variable.
Coefficients
Intercept
Graduation Rate
Student/Faculty Ratio 1 80 20
POINT FORECAST
16.43
Use this model.
Plug and Chug.
The best model includes Grad Rate and SFR (% classes <20 not needed)

20. The university with the most negative residual.
5. Of the 48 universities in the data set, which one has the most surprisingly low alumni giving rate? [10 points]
The university with the most negative residual.
Use the best model, ask for residuals, find the minimum.
MICHIGAN!

21. ANOVA or Regression of SFR on 2 dummies.
6. Bo notices that some of the 48 have “university” in their names, some have “college” and the rest have “institute”. Bo wonders whether these names are predictive of student/faculty ratio? (Formulate and test a relevant hypothesis.) [25 points]
Three groups (p=3)
ANOVA or Regression of SFR on 2 dummies.
SUMMARY OUTPUT
ANOVA df SS MS F
Significance F
Regression 2
2.3290
0.1090
Residual 45
Total 47
Coefficients
Standard Error
t Stat
P-value
Intercept
0.7114
0.0000
Dcollege
3.4120
0.9156
Dinstitute
0.0363

22. Get Ready….. More practice problems (answers) on website.
I’ll host Sunday night Office Hours.
I am available Monday and Tuesday until 2pm.
Check the website to see where I am…you are welcome to join us.