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November 30th, 2015, 06:21 AM  #11 
Senior Member Joined: Oct 2013 From: New York, USA Posts: 661 Thanks: 87  
November 30th, 2015, 05:22 PM  #12 
Senior Member Joined: Dec 2012 From: Hong Kong Posts: 853 Thanks: 311 Math Focus: Stochastic processes, statistical inference, data mining, computational linguistics  In that case, complete the square: $\displaystyle \begin{align}\hat{p}(1\hat{p}) &= \hat{p}  \hat{p}^2 \\ &= (\hat{p}^2  \hat{p}) \\ &= (\hat{p}^2  2\hat{p} \cdot \frac{1}{2} + (\frac{1}{2})^2)  ((\frac{1}{2})^2) \\ &= (\hat{p}  0.5)^2 + 0.25 \end{align} $ Then you can see that the expression is maximised when $\displaystyle \hat{p} = 0.5$. 

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confidence, interval, proportion, sample, size 
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