Discussion
Let \(p\) be the observed proportion of units with the attribute A in a SRSWOR of size \(n\) from a finite population size N. Let P be the population proportion and Q = 1 - P. Then, Var(\(p\)) equals :
Answer: A
\(\mathrm{Var}(p) = \) \(\dfrac{PQ}{n}\dfrac{N - n}{N - 1}\).
A sample proportion is the mean of 0-1 values whose population variance (with divisor N) is PQ. Under SRSWOR the variance of a sample mean is \(\frac{\sigma^{2}}{n}\frac{N - n}{N - 1}\), so
\[ \mathrm{Var}(p) = \frac{PQ}{n} \cdot \frac{N - n}{N - 1} \]
The factor \((N - n)/(N - 1)\) is the finite population correction. \(PQ/n\) is the with-replacement variance, and the other two options use the wrong factor or denominator.
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