3. a) Ratio estimate:
pˆ r  yˆ r
M  y

M
i
iS
iS
i
i
 M  pˆ

M
i
iS
iS
i
i

28 
11
11
9
 30   26 
14
15
13  0.7381  0.74  74%
28  30  26
95% confidence interval:
1 
n  s2
1
Vˆ yˆ r  2  1    r 
M  N  n n  N
 
s r2 
 M
iS
i
 y i  M i  yˆ r


m
M i2  1  i

iS
 Mi
 si2 
  
 mi 
2

n 1
2
2
2
11
11
9

 
 

 28   28  0.7381   30   30  0.7381   26   26  0.7381
14
15
13
 
 
  1.6077

2
2
mi
s
pˆ  1  pˆ i 
si2 
 pˆ i  1  pˆ i   i  i
mi  1
mi
mi  1
1
mi
 1  0.5  0.5
Mi
M  28  30  26 3  28

11 14  1  11 14 
1 
3  1.6077 2
1  2
Vˆ yˆ r  2  1   

  28  0.5 
3
3  32 
13
28  32 
 
 30 2  0.5 
11 15  1  11 15  26 2  0.5  9 13  1  9 13   0.001227
14
 95% confidence interval
0.74  1.96  0.001227  0.74  0.07  0.67 , 0.81
12


b) PPS-sampling:
1
1  11 11 9 
pˆ i        0.7371  0.74  74%

n iS
3  14 15 13 
1
2
Vˆ  pˆ pps  
   pˆ i  pˆ pps  
n  n  1 iS
pˆ pps 
2
2
2
1  11
  11
 9
 

   0.7371    0.7371    0.7371  
3  2  14
  15
  13
 
 0.000731
 95% confidence interval
0.74  1.96  0.000731  0.74  0.05  0.69 , 0.79 
c) Increase in precision, i.e. the standard error of the point estimate becomes smaller
The Horvitz-Thompson estimator should be used
4. a) Estimation in domain:
 yi
tˆyd  N  u  N 
iS d
n
 N
nd  y d
118  38
4484
 22300 
 22300 
 249983
n
118  196  86
400
 
n  su2

ˆ
SE t yd  N   1   
N n

2


 


y 

 i 
nd  y d 2 
1 
1 
 iSd  
2
2
2
2


su 
  yi 

  n  1   nd  1  s yd  nd  y d 

n  1  iS
n
n


 d







118  382 
1 

 117  10 2  118  38 2 
 330.39
399 
400 
 
400  330.39

 SE tˆyd  22300   1 
 20084

22300  400

 99% confidence interval: 249983  2.576  20084  249983  51736  198247 , 301719
b) SRS makes the weighing simple:
y wc  
c
nc
151
237
112
 y cR 
 38 
 43 
 49  42.8
n
500
500
500
c) We need to assume that the nonresponse is MAR and that it may depend on to which
faculty the respondent belongs. The nonresponse must not depend on the question put,
which may be unrealistic in this case. People tend to avoid telling how much money they
spend on things (even if lunch is fairly non-controversial).
d) Two-phase sampling:
n
n
yˆ  R  y R  M  y M
n
n
2
2
n R  1 s R2 n M  1 s M2
n
1  nR
ˆ
ˆ
V y 
 


   y R  yˆ  M  y M  yˆ 
n 1 n
n 1   n n 1  n
n






  20 100  0.2
118  38  196  43  86  49
 42.8
400
n R  1  s R2    yi  y R 2   yi2  n R  y R2 
yR 
iS R

 n

 1  s
iS R

y
iS R , A

2
i

y
iS R , B
2
i

y
iS R , C
2
i
 n R  y R2
 n RA  1  s R2 A  n RA  y R2 A  n RB  1  s R2 B  n RB  y R2B 
RC
2
RC
 n RC  y R2C  n R  y R2 
 117  10 2  118  38 2  195  8 2  196  43 2  85  13 2  86  49 2  400  42.8 2  45091
45091
 113.01
399
400
100
 yˆ 
 42.8 
 37  41.6
500
500
399 113.01 99
112
1  400
100
2
2
Vˆ yˆ 





 42.8  41.6  
 37  41.6   
499 500
499 0.2  500 499  500
500

 0.4316
 s R2 

 99% confidence interval: 41.6  2.576  0.4316  41.6  1.7  39.9 , 43.3