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NSW Y12 Maths - Extension 1 Binomial Distribution Normal Approximation of the Sample Proportion

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Normal Approximation of the Sample Proportion Theory

The sample proportion \(\hat{p}=\dfrac{x}{n}\) is the best estimate of the population proportion from a single sample.\\ The expected value \(E(\hat{P})=\hat{P}\) and the standard deviation \(S D=\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}=S(\hat{p})\)\\ \textbf{CLT (Central Limit Theorem)}\\ The following conditions must be met for CLT to apply:\\ - \(n \hat{p} \geqslant 5\)\\ - \(n(1-\widehat{p}) \geqslant 5\).\\ \textbf{Example 1}\\ Given the proportion elements \(x=240\) and \(n=1000\), calculate the expected value and the standard deviation (or standard error)\\ \textbf{Example 1 solution}\\ $\begin{aligned} E(\hat{p})=\hat{p}=\frac{x}{n}=\frac{240}{1000} & =0.24 \\ SD=s(\hat{p})=\sqrt{\frac{\hat{p}(1-\hat{p})}{n}} & =\sqrt{\frac{0.24 \times 0.76}{1000}} \\ & =0.0135 \end{aligned}$\\ \textbf{Example 2}\\ Does \(x=240\) and \(n=1000\) apply to the CLT?\\ \textbf{Example 2 solution}\\ $\begin{aligned} 1000 \times 0.24 & =240,\quad 240>5 \\ 1000 \times 0.76 & =760,\quad 760>5 . \end{aligned}$\\ \(\therefore\) the proportion elements apply.\\

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