Introduction to Regression (Dr. Monticino). Assignment Sheet Math 1680 Read Chapter 9 and 10 ...
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Transcript of Introduction to Regression (Dr. Monticino). Assignment Sheet Math 1680 Read Chapter 9 and 10 ...
Introductionto
Regression(Dr. Monticino)
Assignment Sheet Math 1680
Read Chapter 9 and 10Assignment # 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)
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Regression LineThe 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
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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)
)()(
)()(
)()()(
)(
)()(
)(
)(
)(
)(
YavgXSD
XavgXYSDrY
YavgXavgXXSD
YSDrY
XavgXXSD
YSDrYavgY
XSD
XavgXr
YSD
YavgY
Origins of Regression Line
Regression line is the smoothed version of the graph of averages
Graph of Averages
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SD and Regression Lines
r = .99 Yellow – Regression Line
Purple – SD Line
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SD and Regression Lines
r = .89 Yellow – Regression Line Purple – SD Line
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SD and Regression Lines
r = .75 Yellow – Regression Line
Purple – SD Line
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SD and Regression Lines
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SD and Regression Lines
r = .54 Yellow – Regression Line Purple – SD
Line
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SD and Regression Lines
0102030405060708090
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X
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SD and Regression Lines
r = .12 Yellow – Regression Line
Purple – SD Line
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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
Size of house 1000 1200 1100 1300 1500 1500 Monthly rent 800 900 750 950 1100 970
Regression Example
Draw a scatter plotFind the correlation coefficient between
the size of house and the rent paidGive the equation for the SD line
Graph the SD lineFind 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
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Sq. Feet
Mo
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Regression ExampleSq. feet Rent
1000 8001200 9001100 7501300 9501500 11001500 970
Avg(X) = 1266.67SD of X = 188.56
Avg(Y) = 911.67SD of Y = 114.81
x: standard units y: standard units-1.41 -0.97-0.35 -0.10-0.88 -1.410.18 0.331.24 1.641.24 0.51
x*y (standard units)1.3760.0361.2450.0592.0300.629
r = 0.896
SD Line
SD line passes through the point (x-average,y-average) and has slope (+/- ) (SD of y)/(SD of x)
00.139)61(.
67.911)67.1266)(61(.)61(.
67.911)67.1266)(61(.
67.911)67.1266(56.188
81.114
XY
XY
XY
XY
)())(()(
)(YaveXavgX
XSD
YSDY
SD Line
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)())(()(
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XSD
YSDrY
67.911)67.1266(61.)896(. XY
Regression Line
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PredictionRent 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?
67.911)67.1266(61.)896(. XY
54.984
67.911)67.12661400(61.)896(.
Y
Y
Regression TidbitsRegression effectRegression fallacyTwo Regression lines
SD and Regression Lines
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)()(
)()( Yavg
XSD
XavgXYSDrY
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
)()(
)()( Yavg
XSD
XavgXYSDrY
Two Regression LinesExample
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
