Chapter 6 (cont.)Difference Estimation
Recall the RegressionEstimation Procedure
2
The Model
The first order linear model
y = response variablex = explanatory variableb0 = y-interceptb1 = slope of the linee = error variable
3
xy 10
x
y
b0Run
Rise b1 = Rise/Run
0 and 1 are unknown populationparameters, therefore are estimated from the data.
The Least Squares (Regression) Line
4
20 1
1
ˆdetermine and to minimize ( ) .n
i ii
b b SSE y y
A good line is one that minimizes the sum
ˆof squared differences ( ) errors
between the scatterplot points and the line.i iy y
1 1 2 2( , ), ( , ), , ( , )n nx y x y x y
0 1ˆi iy b b x
The Least Squares (Regression) Line
5
3
3
ww
w
w
41
1
4
(1,2)
2
2
(2,4)
(3,1.5)
Sum of squared differences = (2 - 1)2 + (4 - 2)2 + (1.5 - 3)2 +
(4,3.2)
(3.2 - 4)2 = 6.89
The smaller the sum of squared differencesthe better the fit of the line to the data.
The Estimated Coefficients
6
To calculate the estimates of the slope and intercept of the least squares line, use the formulas:
1
0 1
2
1
2
1
correlation coefficient
( )
1
( )
1
y
x
n
ii
y
n
ii
x
sb r
s
b y b x
r
y ys
n
x xs
n
The least squares prediction equation that estimates the mean value of y for a particular value of x is:
0 1
1 1
1
ˆ
( )
( )
y b b x
y b x b x
y b x x
Regression estimator of a population mean y
1
1
ˆ ( )
where
ˆEstimated variance of
1ˆ ˆ( ) 12
1
yL x
y
x
yL
yL
y b x
sb r
s
n SSEV
N n n
n MSE
N n
2
1
ˆ( ) .n
i ii
SSE y y
Difference Estimation
In difference estimation, b1 is not calculated.
1ˆ ( )yL xy b x
1
is adjusted up or down by an amount
( ); that is, 1x
y
x b
ˆ ( )
where
yD x
x
y x
d
d y x
Works well when x and y are highly correlated and measured on the same scale.
Difference Estimationˆ ( )
where
yD x
x
y x
d
d y x
Estimated Variance of Difference Estimator
ˆ ( )
where
yD x
x
y x
d
d y x
2
1
( )1ˆ ˆ( ) 1
1
where
n
ii
yD
i i i
d dn
VN n n
d y x
Diff. Est. - example
Achievement Final calculus
Student test score, x grade, y
1 39 65
2 43 78
3 21 52
4 64 82
5 57 92
6 47 89
7 28 73
8 75 98
9 34 56
10 52 75
A math achievement test was given to 486 students prior to entering college. A SRS of n=10 students was selected and their course grades in calculus were obtained. Estimate uy for this population.
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