Topic 4: Statistical Inference
Outline
• Statistical inference
– confidence intervals
– significance tests
• Statistical inference for β1
• Statistical inference for β0
• Tower of Pisa example
Theory for Statistical
Inference
• Xi iid Normal(μ,σ2), parameters unknown
X

X
i
n
s s
2
,
,
s
2
(X


i
 X)
n 1
s
s(X) 
n
2
Theory for Statistical
Inference
X
• Consider variable t 
s (X)
• t is distributed as t(n-1)
• Use distribution in inference for m
– confidence intervals
– significance tests
Confidence Intervals
X  tc s(X)
where tc= t(1-α/2,n-1), the upper (1-a/2)100
percentile of the t distribution with n-1 degrees
of freedom
• 1-a is the confidence level
Confidence Intervals
• X is the sample mean (center of interval)
• s( X ) is the estimated standard
deviation of X, sometimes called the
standard error of the mean
• tc s(X) is the margin of error and
describes the precision of the estimate
Confidence Intervals
• Procedure such that (1-a)100% of the time,
the true mean will be contained in interval
• Do not know whether a single interval is
one that contains the mean or not
• Confidence describes “long-run” behavior
of procedure
• If data non-Normal, procedure only
approximate (central limit theorem)
Significance tests
H 0 :   0 vs H a :   0
t  (X  0 ) s(X)
*
Reject H 0 if t | t c |, t c
*

t(1  α / 2 , n  1)
P  Prob( t  t ), where t ~ t(n - 1)
*
Significance tests
•
•
•
•
•
Under H0 t* will have distribution t(n-1)
P(reject H0 | H0 true) = a (Type I error)
Under Ha, t* will have noncentral t(n-1) dists
P(DNR H0 | Ha true) = b (Type II error)
Type II error related to the power of the test
NOTE
IN THIS COURSE
USE α=.05
UNLESS SPECIFIED
OTHERWISE
Theory for β1 Inference
b1 ~ N (1 ,  (b1 ))
2
where  (b1 )  
2
2
 (X
i
 X)
t  (b1  1 ) / s (b1 )
*
where s(b1 )  s
2
 (X
Under H 0 , t ~ t(n  2)
*
i
 X)
2
2
Confidence Interval for β1
b1 ± tcs(b1)
where tc = t(1-α/2,n-2), the upper (1-α/2)100
percentile of the t distribution with n-2 degrees of
freedom
• 1-α is the confidence level
Significance tests for β1
H 0 : 1  0 vs H a : 1  0
t  (b1  0) s(b1 )
*
Reject H 0 if t | t c |, t c  t(1  α / 2 , n  2)
*
P  Prob( t  t ), where t~t(n  2)
*
Theory for β0 Inference
2
b 0 ~ N ( 0 ,  (b 0 ))
2


1
X
2
2
where  (b 0 )    
2
 n  (X i  X) 
*
t  (b 0   0 ) / s (b 0 )
for s (b 0 ) replace  by s and take
2
Under H 0 , t ~ t(n  2)
*
2
Confidence Interval for β0
b0 ± tcs(b0)
where tc = t(1-α/2,n-2), the upper (1-α/2)100
percentile of the t distribution with n-2
degrees of freedom
• 1-α is the confidence level
Significance tests for β0
H 0 :  0  0 vs H a :  0  0
t  (b0  0) s(b 0 )
*
Reject H 0 if t | t c |, t c  t(1  α / 2 , n  2)
*
P  Prob( t  t ), where t~t(n  2)
*
Notes
• The normality of b0 and b1 follows from
the fact that each of these is a linear
combination of the Yi, each of which is
an independent normal
• For b1 see KNNL p42
• For b0 try this as an exercise
Notes
• Usually the CI and significance test for
β0 is not of interest
• If the ei are not normal but are relatively
symmetric, then the CIs and
significance tests are reasonable
approximations
Notes
• These procedures can easily be modified
to produce one-sided confidence
intervals and significance tests
2
2
2
• Because  (b1 )  
( X i  X ) we
can
make this quantity small by making
n
2
(
X

X
)
large.
 i

i 1
SAS Proc Reg
proc reg data=a1;
model lean=year/clb;
run;
clb option generates confidence intervals
Variable
Intercept
year
Parameter Estimates
Parameter Standard
95% Confidence
DF
Estimate
Error t Value Pr > |t|
Limits
1 -61.12088 25.12982
-2.43 0.0333 -116.43124 -5.81052
1
9.31868 0.30991
30.07 <.0001
8.63656 10.00080
CIs given here….CI for intercept is
uninteresting
Review
• What is the default value of α that we
will use in this class?
• What is the default confidence level
that we use in this class?
• Suppose you could choose the X’s.
How would you choose them if you
wanted a precise estimate of the
slope? intercept? both?
Background Reading
• Chapter 2
– 2.3 : Considerations
• Chapter 16
– 16.10 : Planning sample sizes with power
• Appendix A.6