Answer to Question #184183 in Statistics and Probability for Felicity Sy

Question

Consider a population with values (2, 3, 7, 9).

a. Find the population mean.

b. Find the population variance.

c. Find the population standard deviation.

d. Find all possible samples of size 2 which can be drawn with replacement from this 

population. 

e. Find the mean of the sampling distribution of means.

f. Find the variance of the sampling distribution of means.

g. Find the standard deviation of the sampling distribution of means.

Answer 1

a) population mean

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

=$\frac{2+3+7+9}{4}$

=5.25

b)population variance

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

$=\frac{(2-5.25)^2+(3-5.25)^2+(7-5.25)^2+(9-5.25)^2}{4}$

=8.1875

c)population standard deviation

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

=$\sqrt{8.1875}$

=2.8614

d) possible sample of size 2.

There are 16 possible samples that can be drawn with replacement.

(2,2) (2,3) (2,7) (2,9)

(3,2) (3,3) (3,7) (3,9)

(7,2) (7,3) (7,7) (7,9)

(9,2) (9,3) (9,7) (9,9)

e)mean of the sampling distribution of means.

$\mu_{\bar x}=\frac{\sum \bar x_i}{n}$

=$\frac{84}{16}$

=5.25

f) variance of the sampling distribution of means.

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

=$\frac{65.5}{16}$

=4.09375

g)standard deviation of the sampling distribution of means.

${\sigma_ {\bar x_i} }=\sqrt{\sigma_ {\bar x_i} ^2}$

$=\sqrt{4.09375}$

=2.0233