Question

Random samples of size n = 2 are drawn from a finite population that consists of the numbers 2, 4, 6, and 8. (a) Calculate the mean and the standard deviation of this population. (b) List the six possible random samples of size n = 2 that can be drawn from this population and calculate their means. (c) Use the results of part (b) to construct the sampling distribution of the mean for random samples of size n = 2 from the given population. (d) Calculate the standard deviation of the sampling distribution obtained in part (c) and verify the result by substituting n = 2, N = 4, and the value of o obtained in part (a) into the second of the two standard error formulas on page 275.

Answer 1

Answer:

Given that,

$\rightarrow$ Random samples of size n = 2 are drawn from a finite population that consists of the numbers 2,4,6, and 8.

(a).

Calculate the mean and the standard deviation of this population:

$\rightarrow$ We have population: 2, 4, 6, 8

Mean ($\mu$)=(2+4+6+8)/4

=20/4

$\mu$=5

Standard deviation($\sigma$):

We know that,

σ? Σε – μ)? η –1

12 - (2-5) + (4-5) + (6-5) + (8-5) 4-1

$\sigma^2=\frac{ 9+1+1+9}{3}$

$\sigma^2=\frac{ 20}{3}$

$\sigma^2=6.6667$

= V6.6667

$\sigma=2.5819$

(b).

List the six possible random samples of size n = 2 that can be drawn from this population and calculate their means:

Samples are given by:

Samples	Mean
(2,4)	3
(2,6)	4
(2,8)	5
(4,6)	5
(4,8)	6
(6,8)	7

(c).

Use the results of part (b) to construct the sampling distribution of the mean for random samples of size n =2 from the given population:

Sampling distribution of the mean is:

$\mu _{\bar{x}}=\mu =5$

(d).

Calculate the standard deviation of the sampling distribution obtained in part c) and verify the result by substituting n = 2, N = 4, and the value of σ obtained in part (a):

We know that,

$\sigma _{\bar{x}}=\frac{\sigma }{\sqrt{n}}$

$=\frac{2.5819 }{\sqrt{5}}$

=2.5819/2.2361

=1.1546

For n=2:

$\sigma _{\bar{x}}=\frac{\sigma }{\sqrt{n}}\times \sqrt{\frac{N-n}{N-1}}$

$=\frac{2.5819 }{\sqrt{2}}\times \sqrt{\frac{4-2}{4-1}}$

$=\frac{2.5819 }{1.4142}\times \sqrt{0.6667}$

$=1.8257\times 0.8165$

$\sigma _{\bar{x}}$ =1.4907 (Approximately)