When Sampling Without Replacement from a Finite Population of Size $$\sigma _ { x } ^ { - } = \frac { \sigma } { \sqrt { n } } \sqrt { \frac { N - n } { N - 1 } }$$

Essay

When sampling without replacement from a finite population of size N, the following formula is used to find the standard deviation of the population of sample means: $$\sigma _ { x } ^ { - } = \frac { \sigma } { \sqrt { n } } \sqrt { \frac { N - n } { N - 1 } }$$
However, when the sample size $\mathrm { n }$ , is smaller than $5 \%$ of the population size, $\mathrm { N }$ , the finite population correction $\mathrm { f }$ , $\sqrt { \frac { \mathrm { N } - \mathrm { n } } { \mathrm { N } - 1 } }$ , can be omitted. Explain in your own words why this is reasonable. For $\mathrm { N } = 200$ , find the values of the finite population correction factor when the sample size is $10 \% , 5 \% , 3 \% , 1 \%$ of the population, respectively. What do you notice?