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Question 2.3
Given:
? 
?
? ,
?
? 
?
? 
?
? 
?
?
? 
?
?
? 
?
a) The expected value of the sample size is:
?
?
?
?
?
?
?
? 
?
?
?
? ?
?
?
?
?
?
?
?
?
? 
?
?
? 
?
?
?
?
?
? ?? ? ?? ? ??
?
?
?
? 
b) The variance of the sample size is:
?
?
?
?
?
?
?
? 
?
?
?
? ?
?
?
?
?
?
?
?
?
?
 ? 
?
?
? 
?
?
?
 ? 
?
?
? 
?
?
?
 ? 
?
?
?
?
?
? 
?

??
 ? 
?
? 
?

??
 ? 
?
? ???? ?
?
?
?
?  ?  ? 
?
?
?
? 
c) Assuming
?
is constant for all , that is
?
? ,
?
?
?
? 
The value of so that ?
?
? is same as in case (a) is computed as follows:
 ? 
?


? 
Therefore the constant
?
? 
The new variance is:
?
?
?
? 
?
 ? 
?
? 
?

??
 ? 
?
?
?
?
? 
The new variance, 141.1 is greater than the variance of 97 in case (a).

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Solution to Question 2.4
a
?
?
?
?
 
?
? ?
?
?
1
250


? 
0.3125
0.3125
2
350


? 
0.8750
0.0000
3
150


? 
0.5625
0.5625
4
50


? 
0.2500
1.0000
Total
800
1.0000
? ? ?   ?
??
?
? ?
?
? ?
?
?
? 
Average cluster size ? ? . Therefore, expected number of individuals to be interviewed is:
?
?
?
?   ? ?  ?  ? 
?
?
?
?
?
?
?
?
??
?
? 
?
?

?

?
  ? 

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Solution to Question 2.5
?
0.4
0.3
0.2
0.1
Solution 3(a)
Computation of
?
:
?
?
?
?
?
? 
?
?
?
? 
?
?
?
?  ?  ?  ? 
?
?
?
?
?
? 
?
?
?
? 
?
?
?
?  ?  ?  ? 
?
?
?
?
?
? 
?
?
?
? 
?
?
?
?  ?  ?  ? 
Computation of
??
:
??
?
?
? 
??
?
?
?
?
? 
?
?
?
?  ?  ? 
??
?
?
?
?
? 
?
?
?
?  ?  ? 
Computation of
??
:
??
?
??
? 
??
?
?
? 
??
?
?
?
?
? 
?
?
?
?  ?  ? 
Computation of
??
:
??
?
??
? 
??
?
??
? 
??
?
?
? 
Solution to 3(b)
i. Direct calculation of ?
?
? from the definition
?
?
?
?
?
??
?
?
???
?
?
??
?
?
?
??
?
?
?
??
?
?
?
?
?
 ?  ? 
?
?
?
?
?
ii. Calculation of ?
?
? from
?
?
?
?
?
?
?
???
?
?
?
?
?
?
???
?
?
? 
?
? 
?
?  ?  ? 
?
?
?
? 

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Unformatted Attachment Preview

Question 2.3Given: = 10001 = 600, = 0.1, 12 = 300 = 0.1, 23 = 100 = 0.8, 3a) The expected value of the sample size is:( ) = (1 ) + (2 ) + (3 )( ) = 1 1 + 2 2 + 3 3( ) = 600(0.1) + 300(0.1) + 100(0.8)( ) = 170b) The variance of the sample size is:( ) = (1 ) + (2 ) + (3 )( ) = 1 1 (1 1 ) + 2 2 (1 2 ) + 3 3 (1 3 )( ) = 600(0.1)(1 0.1) + 300(0.1)(1 0.1) + 100(0.8)(1 0.8)( ) = 54 + 27 + 16( ) = 97c) Assuming is constant for all , that is = ,( ) = The value of so that ( ) is same as in ...
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