Question

Consider a population consisting of 1,2,3,4 and 5. Supposed samples of size 3 are drawn from this population. describe the sampling distribution of the sample means.

• What is the mean and variance of the sampling distribution of the sample means?

Answer 1

Population size $N=5$

Sample size $n=3$

Possible samples

$=C_5^3= \binom{5}{3}=10$ samples

$\def\arraystretch{1.5}\begin{array}{c:c:c}no &Sample& Sample mean\\\hline1&1,2,3&2.0\\\hdashline2&1,2,4&2.33\\\hdashline3&1,2,5&2.67\\\hdashline4&1,3,4&2.67\\\hdashline5&1,3,5&3.0\\\hdashline6&1,4,5&3.33\\\hdashline7&2,3,4&3.0\\\hdashline8&2,3,5&3.33\\\hdashline9&2,4,5&3.67\\\hdashline10&3,4,5&4.0\\\hline\end{array}$

$\def\arraystretch{1.5}\begin{array}{c:c:c} Sample mean&X_i-\bar X&(X_i-\bar X)^2(\\\hline2.0&-1.0&1.0\\\hdashline2.33&-0.67&0.44\\\hdashline2.67&-0.33&0.11\\\hdashline2.67&-0.33&0.11\\\hdashline3.0&0&0\\\hdashline3.33&0.33&0.11\\\hdashline3.0&0&0\\\hdashline3.33&0.33&0.11\\\hdashline3.67&0.67&0.44\\\hdashline4.0&1.0&1.0\\\hline\end{array}$

Mean $\mu=\dfrac{\sum X_i}{N}=\dfrac{30}{10}=3.0$

Variance $\sigma^2=\dfrac{\sum(X_i-\bar X)^2}{N-1}$

$\sigma^2 =\dfrac{3.21}{9}=0 .357$

Sample means probability distribution