And Review Linear Regression Analysis We have been comparing the association between two quantitative variables: House Price and Size Burger Protein and Fat Content Fuel Economy (miles per gallon) and horsepower What Weve Learned What We Have Learned R is always between -1 and 1 Because of that, each predicted Y is fewer SDs away from its mean than the corresponding X was This is called regression to the mean Residuals Residuals tell us how well a model works If we plot residuals against predicted values we look for a boring, random graph. If we see a pattern we must re-examine the data to see why. Square the correlation coefficient This number tells us what fraction of the variation of the response is accounted for by the regression model It is a measure of how successful the regression is in linearly relating y to x. Example: Lets relate the SIZE of a house to the PRICE of a house. Example 90,000 square feet the size of 50 average sized family homes Just Checking B) Is the correlation of Price and Size positive or negative? How do you know? Answer: Its positive. The correlation and the slope have the same sign. A final price tag of 75 million dollars Just Checking 30 bedrooms and 20 bathrooms Just Checking You find that your house in Saratoga is worth $100,000 more than the regression model predicts. Should you be very surprised? Answer: No, the standard deviation of the residuals is thousand dollars. We shouldnt be surprised by any residual smaller than 2 standard deviations and a residual of $100,000 is less than 2*53, car garage, three swimming pools The amenities don't finish there. Also constructed is an adult movie theater with a balcony, four fireplaces, a formal dining room that seats 30, all 23 full baths with full-sized Jacuzzis, 160 tripled paned windows and Brazilian mahogany French-style doors that alone cost $4 million. The banquet kitchen features two large commercial gas stoves, four commercial built-in refrigerators and a Japanese-style steakhouse island that seats 12. R Squared But how big should R- Squared be? Data from scientific experiments: 80-90% Data from surveys Much lower! 50-30% President of a financial services company reports that although his regressions give R-Squared below 2%, they are highly successful because those used by his competitors are much lower! Homework on Residuals Pg 193, # 11, 23, 27, 33, 45