Question

Problem: A sample with size n = 4 is drawn from the set 5, 6, 9, 12a 13. Construct the sampling distribution of the means. Find the mean and the standard deviation. Include your solution and box your final answer in mu overline x and 2 sigma overline x

Answer 1

Solution

Population size $N= 5$

Sample size $n=4$

Possible samples

$=C_5^4= \binom{5}{4}=5$ Samples

$\begin {matrix}no.& Sample &Mean\\1.&5,6,9,12&8.00\\2.&5,6,9,13&8.25\\3.&5,6,12,13&9.00\\4.&5,9,12,13&9.75\\5.&6,9,12,13&10.00\end{matrix}$

1. Mean $=\sum \bar Xf(\bar X)$

$\begin {matrix}\bar X&f&f(\bar X)&\bar Xf(\bar X)&\bar X^2f(\bar X)\\8.00&1&1/5&1.60&12.8000\\8.25&1&1/5&1.65&13.6125\\9.00&1&1/5&1.80&16.2000\\9.75&1&1/5&1.95&19.0125\\10.00&1&1/5&2.00&20.0000\\\sum&5&1&9.00&81.625\end{matrix}$

Mean $=\sum \bar Xf(\bar X)$

Mean $=9.000$

2. Standard deviation

$\sigma=\sqrt {\sum \bar X^2 f(\bar X)-(\sum \bar X f(\bar X))^2 }$

$\sigma=\sqrt{81.625-81.000}$

$\sigma =0.791$