Some Solutions

• a. What is the formula to estimate the sample variance s
2  n
1

1 i n 

1
 y i
 y

2

2c. (a54p10.sas). Do any samples occur where the same patient was selected on each of the selections?
Which ones?
What is the chance (probability) that this will happen?
Is the number of times that the same subject was selected each time equal to what you would expect?
Why or why not.

Do any samples occur where the same patient was selected on each of the selections?
(Note: sid1, sid2, etc are the subject IDs)
PROC PRINT DATA=d NOOBS SPLIT="*";
VAR sample sid1-sid3 ;
TITLE2 "Table 2. List of &nsamp Simple Random Samples ";
RUN ;

Table 2. List of 40 Simple Random Samples
1st 2nd 3rd
ID ID ID sample Sel. Sel. Sel.
etc
1 5 6 6
2 3 3 6
3 2 2 3
4 6 3 1
5 3 3 3 Subject 3 selected each time
6 1 1 4
7 6 5 5
8 4 6 4
9 4 3 6
10 1 6 5
11 5 4 4
12 1 6 5
13 4 2 3
14 4 6 6

Do any samples occur where the same patient was selected on each of the selections? (Add these statements to tabulate distinct samples)
– DATA d1;
– SET d;
– snum= 100 *sid1+ 10 *sid2+sid3;
– RUN ;
– PROC PRINT DATA=d1 NOOBS SPLIT="*";
– VAR sample snum sid1-sid3 ;
– TITLE2 "Table 2a. List of &nsamp Simple Random Samples ";
– RUN ;
– PROC FREQ DATA=d1;
– TABLES snum;
– TITLE2 "Table 2b. Subjects in &nsamp Samples with replacement";
– RUN ;

Table 2b. Subjects in 40 Samples with replacement
The FREQ Procedure
Cumulative Cumulative snum Frequency Percent Frequency Percent
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114 1 2.50 1 2.50
121 1 2.50 2 5.00
143 1 2.50 3 7.50
165 3 7.50 6 15.00
166 1 2.50 7 17.50
213 1 2.50 8 20.00
215 1 2.50 9 22.50
223 2 5.00 11 27.50
231 1 2.50 12 30.00
232 1 2.50 13 32.50
233 1 2.50 14 35.00
236 1 2.50 15 37.50
312 1 2.50 16 40.00
324 1 2.50 17 42.50
333 3 7.50 20 50.00 The three selections are subject 3
336 1 2.50 21 52.50

2c. (a54p10.sas). What is the chance
(probability) that the same patient was selected on each of the selections?
# of samples:
n 
6
3 
216
# of samples with same selection:
111, 222, 333, 444, 555, 666
6
Pr ( subject)=
216

0.0277

2c. (a54p10.sas). Is the number of times that the same subject was selected each time equal to what you would expect?
# of samples expected:
  
1.11
No- we observed 3, and only expected 1.11 samples.
This may be because we only took 40 samples.

• a. If we count samples as different if either the subjects included are different or if the order of selection of subjects is different, how many different possible sample could we select?
Note that N=6, n=3
Sample
N N N
______ _______ ______
Position 1 Position 2 Position 3
N n

6
3

216
This counts the samples as different for all subjects selected in all orders.

3b. How many times would you expect that each sample would occur if 4320 samples were selected?
4320

1
216

 
 
20

• g. Suppose that three different subjects are selected. In how many different orders can 3 different subjects be selected (this is the number of permutations).
ID=1, ID=2, ID=3
Sample n n-1 n-2
______ _______ ______
Position 1 Position 2 Position 3 n !
  
1
 n
    
3! 6

When Sampling Without Replacement:
3a. If we count samples as different if either the subjects included are different, or if the order of selection of subjects is different, how many different possible samples could we select?
• N=6, n=3
Sample
N N-1 N-2
______ _______ ______
Position 1 Position 2 Position 3

N !
N
 n

!

     
   
    
120

When Sampling Without Replacement:
3b. How many times would you expect that each sample would occur if 4320 samples were selected?
4320

120
216

 
 
  
2340

When Sampling Without Replacement:
3b. In how many different orders can the same three subjects be selected? (This is the number of permutations.)
     n !