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p_{1}-p_{2}

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4.2.1 Sample distribution of the difference of two proportions

We must check two conditions before applying the normal model to $\hat{p}_{1}-\hat{p}_{2}$ . First, the sampling distribution for each sample proportion must be nearly normal, and secondly, the samples must be independent. Under these two conditions, the sampling distribution of $\hat{p}_{1}-\hat{p}_{2}$ may be well approximated using the normal model.

Conditions for the sampling distribution of $\hat{p}_{1}-\hat{p}_{2}$ to be normal The difference $\hat{p}_{1}-\hat{p}_{2}$ tends to follow a normal model when • each proportion separately follows a normal model, and • the two samples are independent of each other. The standard error of the difference in sample proportions is $\displaystyle SE_{\hat{p}_{1}-\hat{p}_{2}}=\sqrt{SE_{\hat{p}_{1}}^{2}+SE_{\hat% {p}_{2}}^{2}}=\sqrt{\frac{p_{1}(1-p_{1})}{n_{1}}+\frac{p_{2}(1-p_{2})}{n_{2}}}$ (4.2) where $p_{1}$ and $p_{2}$ represent the population proportions, and $n_{1}$ and $n_{2}$ represent the sample sizes.

For the difference in two means, the standard error formula took the following form:

\displaystyle SE_{\bar{x}_{1}-\bar{x}_{2}}=\sqrt{SE_{\bar{x}_{1}}^{2}+SE_{\bar% {x}_{2}}^{2}}

The standard error for the difference in two proportions takes a similar form. The reasons behind this similarity are rooted in the probability theory of Math104, which is described for this context in Exercise 3.2.3 from the extra examples.