Page 291 - DMTH404_STATISTICS

Unit 21: Confidence Intervals

which is the same as Notes





1

P{ b (X Y) (   ) a}   
1 2
or
 
 

 b (X Y) (   ) a 

P   1 2    1  
    1  1     1  1     1  1  
 n m   n m   n  
    m  
1 n 1 m
Here X   X and Y   Y
n 1 i m 1 i
(X Y) (   )


Since 1 2 ~ N(0,1).
 1 1 
   
 n m 
we can choose a and b to satisfy
 
 
  b (X Y) (   ) a 


P   1 2    1  
    1  1     1  1     1  1  
 n m   n m   n  
    m  
provided that  is known. There are infinitely many such pairs of values (a, b). In particular, an
   1 1  1/2  
intuitively reasonable choice is a = b = c, say. In that case c/ s       Z  /2 and the
   n m   
confidence interval is

   1 1  1/2  1 1  1/2  


 (X Y) –     Z  /2 ,(X Y)     Z  /2
   n m   n m   
 1 1  1/2
The length of the intaval is 2    Z . Given  and  one can choose n and m to get a
/2
 n m 
desired length confidence interval.
2
If  is unknown, we have from
P{– b < (X – Y) (   2 ) a}   


1
1
that
 
 


  b (X Y) (   ) a 
P   1 2    1  
 S   1  1  S   1  1  S   1  1  


  n m    n m    n m  


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