Question

Solve the given problems.

1. A population consists of the four numbers 1, 2, 4 and 5. List all the possible

samples of size n = 3 which can be drawn with replacement from the population.

Find the following:

a. Population mean

b. Population variance

c. Population standard deviation.

d. Mean of the sampling distribution of sample means

e. Variance of the sampling distribution of sample means

f. Standard deviation of the sampling distribution of sample means

Answer 1

a) population mean

$\mu=\frac{\sum x_i}{N}$

$=\frac{1+2+4+5}{4}$

=3

b) population variance

$\sigma^2=\frac{\sum (x_i-\mu)}{N}$

$=\frac{(1-3)^2+(2-3)^2+(4-3)^2+(5-3)^2}{4}$

=2.5

c)Population standard deviation

$\sigma=\sqrt {\sigma^2}$

$=\sqrt {2.5}$

=1.58

d)  Mean of the sampling distribution of sample means

$\mu=\frac{\sum \bar x_i}{n}$

$=\frac {12}{4}$

=3

e) Variance of the sampling distribution of sample means

$variance=\frac{\sum (\bar x_i-\mu)^2}{n-1}$

$=\frac{1.111}{4-1}$

=0.3704

f) Standard deviation of the sampling distribution of sample means

$\text{standard deviation}=\sqrt {variance}$

=$\sqrt{0.3704}$

=0.6086