My Math Forum sample size for confidence interval of proportion

November 30th, 2015, 06:21 AM   #11
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Quote:
 Originally Posted by tttim I haven't learnt differentiation
The derivative of x^w where x is a variable and w is a constant is wx^(w-1). The simplest example is that the derivative of x^2 is 2x.

November 30th, 2015, 05:22 PM   #12
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Quote:
 Originally Posted by tttim I haven't learnt differentiation
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 $\displaystyle \hat{p} = 0.5$.

 Tags confidence, interval, proportion, sample, size

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