9.1 Simple Random Sampling and Point Estimate
Simple random sampling from finite population:
A simple random sample of size n from a finite population of size N is
a sample selected such that each possible sample of size n has the
same probability of being selected.
N
N!
  
. Thus,
 n  n!N  n !
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the probability of a specific random sample being selected is
.
N
 
n
Note: the total number of random samples is
Note: the above sampling is also called a random sample without
replacement since we did not place a selected element back into the
population. That is, the sample element can not be selected twice.
Simple random sampling from infinite population:
 Each element selected comes from the same population.
 Each element is selected independently.
Example:
Suppose we randomly select 30 managers as a sample. Let x1 , x2 ,, x30 be the
annual salaries of the 30 managers (a sample). Suppose 19 of them have completed
the training program. Thus,
30
x
x
i 1
30
i
 51814  sample mean
and
p
Note:
19
 0.63  sample proportion
30
x and p are sensible estimates of   51800 and p  0.6 .
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Point estimate:
A point estimate is a statistic based on a sample of size n (not
necessary to be simple random sample) from a finite population of
size N. Suppose x1 , x2 ,, xn
is the sample. Then,
n
x
x
i 1
n
n
s2 
 x
i 1
i
i
 sample mean
 x
2
 sample variance
n 1
and the sample proportion p are point estimates of the population
mean  , the population variance  2 , and the population proportion
p, respectively.
Note:
x , s 2 and p are not the only estimates. They are just some
sensible estimates.
Example:
Suppose we have a population of 40 elements
148
148
149
149
153
154
155
155
156
156
157
157
158
158
158
158
158
159
159
160
160
160
161
162
162
162
163
163
163
163
164
164
164
164
165
165
165
165
165
166
Suppose the first row of the table of random number is
63271 59986 71744 51102 15141 80714 58683 93108 13554 79945
Please use simple random sampling to obtain a sample of 5 elements.
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Online Exercise:
Exercise 9.1.1
Exercise 9.1.2
Exercise 9.1.3
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