Page 290 - DMTH404_STATISTICS

Statistics

Notes so that

 S 2 1 


1
P b  2  a   1  

  
It is known that
(n – 1)S / ~  2
2
2

n 1
We can therefore choose pairs of intervals (a, b) from the tables of the chi-square distribution. In
particular we can choose a, b so that
 S 2 1  S 2 1 
P  2     /2  P  2  . 
  a    b 
n 1 n 1


Then  x 2 n 1, /2 and  x 2 n 1,1 /2 and the 1 –  level confidence interval for  when  is
2



a b
unknown is
 (n 1)S 2 (n 1)S 
2


 , 
  2  2 
 n 1, /2 n 1,1 /2 



n
2
If however,  is known then (n – 1) S is replaced by  (X – ) and the degrees of freedom of

2
i
1
n
2
2
 is n instead of n – 1, for  (X   ) / 2 ~ c .
2
n
i
1
Figure 21.3 : Chi-square values such that area 1 – /2 and /2 are to their right.

Example 6: Let X , . . . , X and Y , . . . , Y denote respectively independent random
1 2 1 m
samples from the two independent distributions having respectively the probability density
2
2
functions N( ,  ) and N( ,  ). We wish to obtain a confidence interval for  –  .
1 2  2
Consider the interval {(X – Y) – a, (X – Y) + b}. In order that this is a (1 - ) level confidence
interval, we mbt have

P{(X – Y) – a <  –   (X – Y) + b}  1 – 
1 2

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