Sampling Theory

Sampling distribution of means

We take N samples (without replacement) and want to estimate the distribution of the mean for the full population (Np objects):

\begin{gathered} {\mu _{\bar X}} = \mu \\ {\sigma _{\bar X}} = \frac{\sigma }{{\sqrt N }}\sqrt {\frac{{{N_p} - N}}{{{N_p} - 1}}} \\ \end{gathered}

We take N samples from an infinite population (or finite with replacement) and estimate the distribution of the mean:

\begin{gathered} {\mu _{\bar X}} = \mu \\ {\sigma _{\bar X}} = \frac{\sigma }{{\sqrt N }} \\ \end{gathered}

Sampling distribution of the sums and differences

\begin{gathered} {\mu _{{{\bar X}_1} \pm {{\bar X}_2}}} = {\mu _{{{\bar X}_1}}} \pm {\mu _{{{\bar X}_2}}} \\ {\sigma _{{{\bar X}_1} \pm {{\bar X}_2}}} = \sqrt {{\sigma _{{{\bar X}_1}}} + {\sigma _{{{\bar X}_2}}}} \\ \end{gathered}
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