Introduction to Regression (Dr. Monticino)

Assignment Sheet Math 1680  Read Chapter 9 and 10  Assignment # 7 (Due March 2)  Chapter 9 Exercise Set A: 2, 6, 7, 8; Exercise Set B: 3, 4 Exercise Set C: 1, 2; Exercise Set E: 3, 4, 5  Chapter 10 Exercise Set A: 1, 2, 4, 5; Exercise Set B: 3 Exercise Set C: 1; Exercise Set D: 1, 2 Exercise Set E: 1, 2  Test on March 2 on Chapters 1-5, 8, 9, 10.  Emphasis on problems, concepts covered in class and on quizzes

Regression  Regression is used to express how the independent variable(s) is (are) related to the dependent variable  And, to make predictions about the value of the dependent variable based on knowledge of the value of the independent variable  In particular, regression is used to build a linear model to describe the relationship between the independent and dependent variable

Regression FE score = a + b*(MT score)

Regression Line  The regression line is to a scatter diagram as the average is to a list.  The regression line for y on x estimates the average value of y corresponding to each value of x

Linear Regression Model  Again, the regression line provides a linear model for predicting the value of the dependent variable given the value of the independent variable  If there was no correlation between the variables then a reasonable guess for the value of the dependent variable would be the Ave(Y)  If there was very strong correlation between the variables, say correlation 1, then given a value X = Ave(X) + k*SD(X), then one should guess Y = Ave(Y) + k*SD(Y)…see next slide for details

Linear Regression Model Equation of the Regression Line: ( Notice its relationship to the SD Line )

Origins of Regression Line  Regression line is the smoothed version of the graph of averages

Graph of Averages

SD and Regression Lines  r =.99 Yellow – Regression Line Purple – SD Line

SD and Regression Lines  r =.89 Yellow – Regression Line Purple – SD Line

SD and Regression Lines  r =.75 Yellow – Regression Line Purple – SD Line

SD and Regression Lines  r =.54 Yellow – Regression Line Purple – SD Line

SD and Regression Lines  r =.12 Yellow – Regression Line Purple – SD Line

Regression Example  A Denton consumer welfare group investigated the relationship between the size of houses and the rents paid by tenants. The group collected the following information on the sizes (square feet) of six houses and monthly rents (in dollars) paid by tenant

Regression Example  Draw a scatter plot  Find the correlation coefficient between the size of house and the rent paid  Give the equation for the SD line  Graph the SD line  Find the equation for the regression line  Graph the regression line

Regression Example  Use the regression line model to predict the rent for a 1400 sq. ft. house  Suppose that you do not know the square footage of the home, how much would you expect to pay for rent?

Scatter Plot

Regression Example Avg(X) = SD of X = Avg(Y) = SD of Y =

SD Line  SD line passes through the point (x-average,y-average) and has slope (+/- ) (SD of y)/(SD of x)

SD Line

Regression Line

Prediction  Rent for a 1400 sq. ft. house  Suppose that do not know the square footage of the home, how much would you expect to pay for rent?

Regression Tidbits  Regression effect  Regression fallacy  Two Regression lines

Two Regression Lines  Often there are not clear “cause” and “effect” variables  In such cases, it may be just as reasonable to regress either variable with respect to the other  However, need to be clear which variable is being considered the dependent variable (the value being predicted) and which variable is being considered the independent variable in the regression application

Two Regression Lines  Example  Suppose the correlation between husband’s and wife’s IQs is.6. The average husband IQ is 100 with an SD of 10, the average wife IQ is 105 with an SD of 8  Given a husband with an IQ of 110, use regression to estimate his wife’s IQ  Given a wife with an IQ of 100, use regression to estimate her husband’s IQ (Dr. Monticino)  Chapter 9 Review exercises: 2, 3, 5, 8  Chapter 10 Review Exercises: 1, 2, 3, 5, 7