A population has N units with each response being a categorization as below.

={1with probability p0with probability 1-pYi={1with probability p0with probability 1-p

Suppose you take a random sample of size n and calculate the sample proportion p^. In class (see lesson 6), I said 2=(1)(1)s2n=p^(1p^)(n1)when the response is binary, and hence showed that ^()=(1)(1)(1)Var^(p^)=p^(1p^)(n1)(1nN).

Show that 2=(1)(1)s2n=p^(1p^)(n1) when the response Yi is defined as above. Here 2==1()21s2=i=1n(yiy)2n1 is the sample variance.