Question

A drawer contains 4 capsules numbered 2, 3, 5, and 8. a sample of size 3 is drawn with replacement. what is the standard deviation of x-bar?

Answer 1

The table below summarises the calculation for the standard deviation of x-bar.

$\begin{tabular} {|c|c|c|c|} Drawing (x)& \bar{x} & x-\bar{x} & (x-\bar{x})^2\\[1ex] 2,2,2&2&-2.5&6.25\\ 2,2,3&2.3&-2.2&4.84\\ 2,2,5&3&-1.5&2.25\\ 2,2,8&4&-0.5&0.25\\ 2,3,3&2.7&-1.8&3.24\\ 2,3,5&3.3&-1.2&1.44\\ 2,3,8&4.3&-0.2&0.04\\ 2,5,5&4&-0.5&0.25\\ 2,5,8&5&0.5&0.25\\ 2,8,8&6&1.5&2.25\\ 3,3,3&3&-1.5&2.25\\ 3,3,5&3.7&-0.8&0.64\\ 3,3,8&4.7&0.2&0.04\\ 3,5,5&4.3&-0.2&0.04\\ 3,5,8&5.3&0.8&0.64\\ 3,8,8&6.3&1.8&3.24\\ 5,5,5&5&0.5&0.25\\ 5,5,8&6&1.5&2.25\\ 5,8,8&7&2.5&6.25\\ 8,8,8&8&3.5&12.25\\ \end{tabular}$

$\sum\bar{x}=89.9 \\ \\ \bar{\bar{x}}= \frac{\sum\bar{x}}{n}=\frac{89.9}{20} \approx4.5 \\ \\ \sum(x-\bar{x})^2=30.41 \\ \\ s.d= \sqrt{ \frac{\sum(x-\bar{x})^2}{n} } = \sqrt{ \frac{30.41}{20} } = \sqrt{1.5205} =1.23$

Therefore, the standard deviation of x-bar is approximately 1.23.