Question

Samples of size 2 are taken from the population 1, 2, 3, 4, 5, 6 with replacement. Find
the following.
(i) The mean of population
(ii) S.D of the population
(iii) The mean of the sampling distribution of means
(iv) The S.D of the sampling distribution of means.

Answer 1

For the given values of x (1,2,3,4,5,6)

ANSWER 1)

Population mean $\mu = \frac {\sum x}{N} = \frac {1+2+3+4+5+6}{6} = 3.5$

ANSWER 2)

Standard deviation of the population $\sigma = \sqrt \frac{\sum(x- \mu)^2}{N} = \sqrt \frac {17.5}{6} = 1.7078$

ANSWER 3)

The mean of the sampling distribution of means = Expected value of $\bar x :$

E$(\bar x) = \mu = 3.5$

ANSWER 4)

Given n = 2

Standard distribution of the sample distribution of means :

$\sigma$₍$\bar x$₎ $= \sqrt \frac {N - n}{N-1}* \frac {\sigma}{\sqrt n} = \sqrt \frac {6 - 2}{6-1} * \frac {1.7078}{\sqrt 2}$

$\sigma$₍$\bar x$₎ = 1.0801