Download The Distribution of the Sample Mean

1




The slope 1 is the true average change in the
dependent variable y associated with a 1-unit
increase in x.
The slope of the least squares line b1  ˆ1 gives
an estimate of the true slope
This estimate depends on the values of Y,
which are random.
If we can determine the sampling distribution
of the estimate, we can perform inference for
the true slope.
2

The estimator of the slope is
b1  ˆ1 
  x  x  Y  Y   S
S
 x  x 
i
i
2
i
where
xy
ci   xi  x  / S xx
  ci Yi  Y    ciYi
xx
.
This is a linear combination of normal random
variables Yi , and thus it has a normal
distribution.
3
E   ciYi    ci E Yi    ci  0  1 xi   0  ci 1  ci xi
 0  1  ci  xi  x   1
V   ciY    c V Yi  
2
i
  xi  x 
S
2
xx
2
 
2
2
S xx
4

Recall that in simple linear regression we
estimate  2 using SSE  yi2  ˆ0  yi  ˆ1  xi yi

n2
2
s
(which the book calls ).

n2
Then the estimated standard deviation of the
s
ˆ
estimator 1 is sˆ 
.
1
S xx
5

The assumptions of the simple linear
regression model then imply that the
standardized variable
T
ˆ1  1
S ˆ
1

ˆ1  1
S / S xx
has a t distribution with n-2 d.f.
6

1
Confidence intervals and hypothesis tests for
are then carried out in the usual manner.

A 100 1    % confidence interval for 1 is
ˆ1  t /2,n2  sˆ
1
7


Null hypothesis: H 0 : 1  10
Test statistic: t   ˆ1  10  / Sˆ
1

Alternative hypothesis
H a : 1  10
H a : 1  10
H a : 1  10
Rejection region
t  t ,n2
t  t ,n2
t  t /2,n2 or t  t /2,n2
The test H 0 : 1  0 versus H a : 1  0
usefulness of the model.
tests the
8

The sample correlation coefficient r gives a
measure of the linear relationship among X andY.

Whereas for linear regression the X variable is
fixed, here it doesn’t matter which variable is
called X, and which is calledY.

The statistic r is related to the coefficient of
determination in simple linear regression, and
forms an estimate of the population correlation
coefficient  .
9

The sample correlation coefficient for the n
pairs  x1 , y1  , ,  xn , yn  is
r
  x  x  y  y 
 x  x   y  y 
i
2
i

i
i
2

S xy
S xx S yy
Recall that b1  S xy / S xx , so that the estimated
slope and r have the same sign.
10
The value of r is independent of the units in
which x and y are measured
 r lies in the interval 1  r  1
 r = 1 if and only if all  xi , yi  pairs lie on a straight
line with positive slope, and r = -1 if and only if all
pairs lie on a straight line with negative slope.
 The square of the correlation coefficient gives
the value of the coefficient of determination
from fitting the simple linear regression model.
 r measures the degree of the linear relationship

11



Weak .5  r  .5
Moderate .8  r  .5 or .5  r  .8
Strong r  .8 or r  .8

The rationale for calling correlations weak even
when they are as large in absolute value as .5 is
2
r
 .25 , so that if r  .5 the
that even in that case
linear model explains at most 25% of the observed
variation, which is not very impressive.
12

We can think of the pairs  xi , yi  as having been
drawn from a bivariate population of pairs,
with some joint pmf or pdf, and correlation  .

When the joint pdf is bivariate normal, one can
carry out inference for  .

Let (X,Y) be bivariate normal with respective
2
2

,


,

means 1 2 , variances 1 2 , and correlation
coefficient  .
13
If X = x, it can be shown that the (conditional)
distribution ofY is normal with mean
Y x  2  1 2 /  1   2 x /  1 and variance 1   2  22

This fits the simple linear regression model
with 0  2  1 2 /  1 , 1   2 /  1 , and
 2  1   2  22 .

14

The implication is that if the observed pairs
are actually drawn from a bivariate normal
distribution, then the simple linear regression
model is an appropriate way of studying the
behavior ofY given X=x.

If   0 , then Y x  2 , independent of x.
15

When H 0 :   0 is true, the test statistic
T  R n  2 / 1  R 2 has a t distribution with n-2
d.f.

Alternative hypothesis
Ha :   0
Ha :   0
Ha :   0
Rejection region
t  t ,n2
t  t ,n2
t  t /2,n2 or t  t /2,n2
16

The null hypothesis   0 states that there is
no linear relationship between X and Y in the
population.

In the context of regression analysis, we used
ˆ1 / sˆ to test for the absence of a linear
relationship (1  0 ). Since r n  2 / 1  r 2  ˆ1 / sˆ
the tests are equivalent.
1
1
17