3. a) Ratio estimate:
pˆ r  yˆ r 
 M i  yi  M i  pˆ i
iS

 Mi
iS
 Mi
iS

55 
8
19
4
7
 116   19   45 
11
23
4
9  0.8078  0.81  81%
55  116  19  45
iS
95% confidence interval:
1 
n  s2
1
Vˆ yˆ r  2   1    r 
N  n n N
M 
 
 M i  yi  M i  yˆ r 
mi  si2 
2 

M

1

 i  M   m 
i 
i

iS

2
s r2 
iS
n 1

2
2
2
2
8
19
4
7








 116  0.8078    19   19  0.8078    45   45  0.8078 
 55   55  0.8078    116 
11
23
4
9










3
 13.092
pˆ  1  pˆ i 
si2  i
mi  1
1
mi
 1  0.2  0.8
Mi
M  55  116  19  45 4  58.75

 
Vˆ yˆ r 

8 11  1  8 11
4  13.092 2
1

 1 

  55 2  0.8 


2 
200 
3
4  200 
10
58.75 
1
 116 2  0.8 
19 23  1  19 23
22
 19 2  0.8 
4 4  1  4 4
3
 45 2  0.8 
7 9  1  7 9 
8
 0.01623
 95% confidence interval
0.81  1.96  0.001623  0.81  0.25  0.56 ,1 ( Note! The upper limit cannot exceed 1)
 

b) Strategy (ii) is problematic as it would require a pre-inspection of all car parks prior to
sampling. Thus it is not efficient from a cost point-of-view. Strategy (i) is more efficient
in that sense since the car parks can be expected once and for all to get the number of
parking places. However, if strategy (i) should be efficient from an estimation point-ofview we need to ensure that the number of parked cars is generally proportional to the
number of parking places. Such a property is not verified from the data at hand and may
be questioned. Small car parks would have a higher probability to be fully occupied then
have larger parks.
c) There are two possibilities for such a sample:
(i) {Car park 1, Car park 2}={CP1, CP2}
(ii) {Car park 2, Car park 1}={CP2, CP1}
Pr CP1 and CP2 both in the sample  
Pr CP1 is chosen first   Pr CP2 is chosen second | CP1 is chosen first  
 Pr CP2 is chosen first   Pr CP1 is chosen second | CP2 is chosen first  

116 58000
55 58000
55
116



 3.8  10  6
58000 1  55 58000 58000 1  116 58000
4. a) Weighting-class adjusted estimate (simple with SRS):
tˆwc  N  
c
nc
 y cR
n
N  5850  7100  10250  9700  13450  46350

153
221
209
291
 126

tˆwc  46350  
 2250 
 3590 
 3210 
 3420 
 1750  
1000
1000
1000
1000
 1000

 128213830
b) Post-stratified estimate:
y post  N  
c
Nc
 ycR   N c  ycR 
N
c
5850  2250  7100  3590  10250  3210  9700  3420  13450  1750  128265500
c) Cell mean imputation:
Mr Ying, 55 years old  Age class 51-65  44000
Ms Uylan, 24 years old  Age class 18-25  29510
d) Regression imputation:
Fit a regression line: yˆ  b0  b1  x
b1  r 
sy
; b0  y  b1  x
sx
Mr Ying: Age class 51-65. For this class
b1  0.79 
9570
9570
 3500
and b0  44000  0.79 
630
630
 Imputed value = 44000  0.79 
9570
9570
 3500  0.79 
 3670  46148
630
630
Ms Uylan: Age class 18-25. For this class
b1  0.84 
8140
8140
and b0  29510  0.84 
 2310
570
570
 Imputed value = 29510  0.84 
8140
8140
 2310  0.84 
 2140  27471
570
570