Example
NOTE:
the number of different possible samples of n from a population of N elements is given by the formula
N!/(n!(N - n)!) (it may be a BIG number!!!)the definition of a simple random sample implies that each element of the population has an equal probability of being selected; but equal probability of selection of elements is not a sufficient condition for a simple random sample - Q - Why? (See NWW p. 241 bottom.)
Exhibit: (NWW Table 9.1 p. 244) [m8001.gif]
Exhibit: (NWW Figure 9.1 p. 246) [m8002.gif]NOTE: in the social sciences an important example of a population treated as an infinite population is that of the error term in a statistical models, as in a regression model of Y as a function of variables X_{k} and a residual error term; diagnostic procedures for checking randomness may then be used with an estimate of the error term called the residual.
Exhibit: (NWW Figure 9.2 p. 247) [m8003.gif]
Exhibit: (NWW Figure 9.3 p. 248) [m8004.gif]
Parameter | Finite population
with observations X_{1}, X_{2}, ..., X_{N} |
Infinite population
represented by RV X |
Population Mean | m = (S_{i=1 to N}X_{i})/N | m = E{X} |
Population Variance | s^{2} = (S_{i=1 to N}(X_{i} - m)^{2})/N | s^{2} = s^{2}{X} |
Population Standard Deviation | s = (s^{2})^{1/2} | s = (s^{2})^{1/2} |
NOTE: See also box in NWW p. 251 for demonstration that "the population
mean m and variance s^{2}
for a finite population correspond, respectively, to the expected value
and variance of the RV associated with the equal-probability selection
of one population element."