Question

Given the population of numbers 3,6,8,9 and 4.Suppose samples of a size 3 are drawn from this population.

1. What is the mean and the variance of population?

2. how many different samples of size 3 can be drawn from the population? List them with their corresponding mean.

Answer 1

Solution

Population size $N=5$

Sample size $n=3$

1. Mean and Variance of the population

Mean $\mu=\dfrac{\sum X_i}{N}$

$\mu=\dfrac{(3+4+6+8+9)}{5}=6.0$

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

$\def\arraystretch{1.5}\begin{array}{c:c:c}\bar X_i&X_i-\bar X&(X_i-\bar X)^2\\\hline3&-3&9\\\hdashline4&-2&4\\\hdashline6&0&0\\\hdashline8&2&4\\\hdashline9&3&9\\\hline\sum &&26\\\hline\end{array}$

$\sigma^2=\dfrac{26}{5}=5.2$

2. Possible samples and their corresponding means

Possible number of samples

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

The samples and their corresponding means

$\def\arraystretch{1.5}\begin{array}{c:c:c}no &Sample& Sample mean\\\hline1&3,4,6&4.33\\\hdashline2&3,4,8&5.00\\\hdashline3&3,4,9&5.33\\\hdashline4&3,6,8&5.67\\\hdashline5&3,6,9&6.00\\\hdashline6&3,8,9&6.67\\\hdashline7&4,6,8&6.00\\\hdashline8&4,6,9&6.33\\\hdashline9&4,8,9&7.00\\\hdashline10&6,8,9&7.67\\\hline\end{array}$