Page 292 - DMTH404_STATISTICS

Statistics

Notes
n m
 (X  X)2   (Y  Y) 2 (n 1)S  (m 1)S 2
2
i
i


where 1 1  x y



(n m 2) n m 2

It is known that

(X Y) (   2 )

1
~ t n m 2 . We can choose pairs of values (a, b) using Student’s t-distribution


 1 1 
S2   
 n m 
with n + m – 2 degrees of freedom such that
 
 
  b (X Y) (   2 ) a 


1
     1  
 S   1  1  S   1  1  S   1  1  
 n m   n m   n  
    m  
In particular, an intuitively reasonable choice is a = b = c, say. Then
c
 t n m 2, /2



 1 1 
S   
 n m 
   1 1  1/2  1 1  1/2  



and (X Y) S     tn m 2,a/2,(X Y) S    t n m 2, /2 





   n m   n m  
is a 1 –  level confidence interval for  –  .
1 2
Example 7: Let X , . . . , X , and Y , . . . , Y , n, m > 2, denote respectively independent
1 n 1 m
random samples from the two distributions having respectively the probability density fundions
2
N( , 2 ) and N N( , 2 ). We wish to obtain a confidence interval for the ratio  2 / when 
1 1 2 2 2 1 1
and  are unknown.
2
2
2
2
2
Consider the interval (a S /S , bS /S ) a , b > 0, where
2 1 2 1
1 n 1 m
2
2
2
2
S   (X  X) ,S   (Y  Y) ,
1 i 2 i

(n 1) 1 (m 1) 1

1 n 1 m
X   X ,Y   Y . We have
i
i
n 1 m 1
2
 S 2  2 S 
P a 2 2  2 2  b 2 2   1  

 S 1  1 S 1 
so that
2
2
  1 (S /S ) 1 
P   2 2 1 2    1 
  b ( 2 / 1 ) a 
284 LOVELY PROFESSIONAL UNIVERSITY