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.