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Class 28 Get Ready….

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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

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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

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Things I expect you will know

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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) Things I expect you will know

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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. Things I expect you will know

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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. Things I expect you will know

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UNDERSTANDING Coefficient Regression Table Constant Fact Coefficient Regression Table Constant Fact Stars Oh…Fact Movies had fewer Stars!

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Coefficient Regression Table Constant Fact Coefficient Regression Table Constant Fact Stars Oh…Fact Movies had fewer Stars! Secret Formula Regress Y on X1 Regress Y on X1 and X2 Regress X2 on X1

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Secret Formula Coefficient Regression Table Constant Fact Coefficient Regression Table Constant Fact Stars Regress Y on X1 Regress Y on X1 and X2 Regress X2 on X1

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UNDDERSTANDING Coefficient Regression Table Constant Fact Coefficient Regression Table Constant Fact Stars Oh…Fact Movies had fewer Stars!

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UNDERSTANDING Coefficient Regression Table Constant Fact Coefficient Regression Table Constant Fact Stars Fact Movies averaged 0.32 fewer Stars! Secret Formula

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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. Regression is the line through a cloud of points

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Class 28 Assignment VariableSchool Graduation Rate % of Classes Under 20 Student/Faculty Ratio Alumni Giving Rate Description The name of the Universit y 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 Median Mode Standard Deviation Skewness Minimum Maximum Count 48

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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 Coefficients Standard Errort StatP-value Intercept E-06 Graduation Rate E-10

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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.

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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] CoefficientsStandard Errort StatP-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 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.

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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] The best model includes Grad Rate and SFR (% classes <20 not needed) CoefficientsStandard Errort StatP-value Intercept Graduation Rate % of Classes Under Student/Faculty Ratio Coefficients Intercept Graduation Rate Student/Faculty Ratio Intercept1 Graduation Rate80 Student/Faculty Ratio20 POINT FORECAST16.43 Don’t Use this variable. Use this model. Plug and Chug.

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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!

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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 dfSSMSF Significance F Regression Residual Total CoefficientsStandard Errort StatP-value Intercept Dcollege Dinstitute

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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.

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