Sampling distributions - the difference between two proportions

If $X \sim B (n_{x}, p_{x})$ and $Y \sim B (n_{y}, p_{y})$ where $X$ and $Y$ are independent, then the expectation and variance of the distribution $P_{x} - P_{y}$ are given by

E (P_{x} - P_{y}) = p_{x} - p_{y} a n d V a r (P_{x} - P_{y}) = \frac{p_{x} q_{x}}{n_{x}} + \frac{p_{y} q_{y}}{n_{y}}

If $n p$ and $n q$ are both greater than 5 for each population, then $P_{x} - P_{y}$ can be approximated with a normal distribution. In other words

P_{x} - P_{y} \sim N (p_{x} - p_{y}, \frac{p_{x} q_{x}}{n_{x}} + \frac{p_{y} q_{y}}{n_{y}})

the confidence interval is given by

p_{x} - p_{y} \pm c \sqrt{V a r (P_{x} - P_{y})}

The value of $c$ depends on the level of confidence you need for your confidence interval.

Level of confidence	Value of c
90%	1.64
95%	1.96
99%	2.58
Sources: 1

Sampling distributions - the difference between two proportions

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