Sampling distributions - the difference between two means

If $X \sim N (μ_{x}, σ_{x}^{2})$ and $Y \sim N (μ_{y}, σ_{y}^{2})$ where $X$ and $Y$ are independent, then the expectation and variance of the distribution $\bar{X} - \bar{Y}$ are given by

E (\bar{X} - \bar{Y}) = \bar{μ_{x}} - \bar{μ_{y}} a n d V a r (\bar{X} - \bar{Y}) = \frac{σ_{x}^{2}}{n_{x}} + \frac{σ_{y}^{2}}{n_{y}}

If the population variances $σ_{x}^{2}$ and $σ_{y}^{2}$ are known, then $\bar{X} - \bar{Y}$ is distributed normally. In other words

\bar{X} - \bar{Y} \sim N (μ_{x} - μ_{y}, \frac{σ_{x}^{2}}{n_{x}} + \frac{σ_{y}^{2}}{n_{y}})

the confidence interval is given by

\bar{x} - \bar{y} \pm c \sqrt{V a r (\bar{X} - \bar{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
If $σ_{x}^{2}$ and $σ_{y}^{2}$ are unknown, then you will need to approximate them with $s_{x}^{2}$ and $s_{y}^{2}$ . If the samples sizes are large, then you can still use the normal distribution. If the sample sizes are small, then you will need to use the t-distribution instead.

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Sampling distributions - the difference between two means

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