Question

A population consists of the numbers 1, 2, 3, and 4 with the sample size of 2. what is the mean that has a probability of 2/6 or 1/3 or 0.33?

Answer 1

$All\,\,samples:\\\left( 1,2 \right) ,\bar{x}=1.5\\\left( 1,3 \right) ,\bar{x}=2\\\left( 1,4 \right) ,\bar{x}=2.5\\\left( 2,3 \right) ,\bar{x}=2.5\\\left( 2,4 \right) ,\bar{x}=3\\\left( 3,4 \right) ,\bar{x}=3.5\\P\left( \bar{x}=1.5 \right) =P\left( \bar{x}=2 \right) =P\left( \bar{x}=3 \right) =P\left( \bar{x}=3.5 \right) =\frac{1}{6}\\P\left( \bar{x}=2.5 \right) =\frac{1}{3}\Rightarrow \bar{x}=2.5$