HP (Hewlett-Packard) 50g ユーザーズマニュアル

Page 18-52

From which it follows that the standard deviations of x and y, and the 
covariance of x,y are given, respectively, by

 , 

, and  

Also, the sample correlation coefficient is

In terms of 

⎯x, ⎯y, S

xx

, S

yy

, and S

xy

, the solution to the normal equations is:

 ,    

The regression curve of Y on x is defined as Y = 

Α + Β⋅x + ε.  If we have a set 

of n data points (x

i

, y

i

), then we can write  Y

i

 = 

Α + Β⋅x

i

 + 

ε

I

, (i = 1,2,…,n), 

where Y

i

 = independent, normally distributed random variables with mean (

Α + 

Β⋅x

i

) and the common variance 

σ

2

;

ε

i

 = independent, normally distributed 

random variables with mean zero and the common variance 

σ

2

.

Let y

i

 = actual data value, 

^

y

i

 = a + b

⋅x

i

= least-square prediction of the data.  

Then, the prediction error is:  e

i

 = y

i

 - 

^

y

i

 = y

i

 - (a + b

⋅x

i

).

An estimate of 

σ

2

 is the, so-called, standard error of the estimate,

Here are some concepts and equations related to statistical inference for linear 
regression:

1

−

=

n

S

s

xx

x

1

−

=

n

S

s

yy

y

1

−

=

n

S

s

yx

xy

.

yy

xx

xy

xy

S

S

S

r

⋅

=

x

b

y

a

−

=

2

x

xy

xx

xy

s

s

S

S

b

=

=

)

1

(

2

1

2

/

)

(

)]

(

[

2

1

2

2

2

2

1

2

xy

y

xx

xy

yy

i

n

i

i

e

r

s

n

n

n

S

S

S

bx

a

y

n

s

−

⋅

⋅

−

−

=

−

−

=

+

−

−

=

∑

=