Question

A sample with size n = 3 is drawn from the set 5, 6, 8, 12, and 20. Construct the sampling distribution of the sample means.

Answer 1

The population size is $N=5$ and the sample size is $n=3$. The number of possible samples which can be drawn without replacement is

$\binom{N}{n}=\binom{5}{3}= 10$

The 10 combinations and their sample means are given below.

The sample mean is derived from the formula, $\bar{x_i}={\displaystyle\sum^{10}_{i=1} x_i\over 3}$

sample values Sample mean

(5,6,8) ${19\over3}=6.33$

(5,6,12) ${23\over3}=7.67$

(5,6,20) ${31\over3}=10.33$

(5,8,12) ${25\over3}=8.33$

(5,8,20) ${33\over3}=11$

(6,8,12) ${26\over3}=8.67$

(6,8,20) ${34\over3}=11.33$

(8,12,20) ${40\over3}=13.33$

(5,12,20) ${37\over3}=12.33$

(6,12,20) ${38\over3}=12.67$

The sampling distribution for the sample means is given as,

$\bar{x_i}$ 6.33 7.67 10.33 8.33 11 8.67 11.33 13.33 12.33 12.67

$p(\bar{x_i})$ ${1\over10}$ ${1\over10}$ ${1\over10}$ ${1\over10}$ ${1\over10}$ ${1\over10}$ ${1\over10}$ ${1\over10}$ ${1\over10}$ ${1\over10}$