Answer to Question #320006 in Statistics and Probability for Mark

Question

Solve the given problem. Dont forget to show your solutions.

1.AA pop fourulstion consist of the four numbers 1,2,4 and 5. List all the possible samples of sizes n=3 wich 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

The number of the samples is the number of combinations of size n from a set S of size m with replacement:

$N=\begin{pmatrix} n+m-1\\ n \end{pmatrix}=\cfrac{(n+m-1)!}{n!\cdot (m-1)!}=\\ =\cfrac{(4+3-1)!}{3!\cdot (4-1)!}=\cfrac{6!}{3!\cdot 3!}=\cfrac{4\cdot5\cdot6}{2\cdot3}=20.$

All the possible samples of sizes n=3 wich can be drawn with replacement from the population:

$\{ (1,1,1), (1,1,2),(1,1,4),(1,1,5),(1,2,2),\\ (1,2,4),(1,2,5),(1,4,4),(1,4,5),(1,5,5),\\ (2,2,2),(2,2,4),(2,2,5),(2,4,4),(2,4,5),\\ (2,5,5),(4,4,4),(4,4,5),(4,5,5),(5,5,5)\}.$

a. Population mean:

$\mu=\sum x_i\cdot P(x_i)=\\ =1\cdot\cfrac{1}{4}+2\cdot\cfrac{1}{4}+4\cdot\cfrac{1}{4}+5\cdot\cfrac{1}{4}=3.$

b. Population variance:

$\sigma^2=\sum(x_i-\mu)^2\cdot P(x_i),$

$X-\mu=\begin{Bmatrix} 1-3, 2-3, 4-3, 5-3 \end{Bmatrix}=$

$=\begin{Bmatrix} -2, -1, 1, 2 \end{Bmatrix},$

$\sigma^2=(-2)^2\cdot \cfrac{1}{4}+(-1)^2\cdot \cfrac{1}{4}+1^2\cdot \cfrac{1}{4}+2^2\cdot \cfrac{1}{4}=2.5.$

c. Population standard deviation:

$\sigma=\sqrt{2.5}=1.58.$

For d., e., f., we'll use the properties of sampling distributions of sample means.

d. Mean of the sampling distribution of sample means:

$\mu_{\bar x} =\mu=3.$

e. Variance of the sampling distribution of sample means:

$\sigma^2_{\bar x}=\cfrac{\sigma^2}{n}=\cfrac{2.5}{2}=1.25.$

f. standard deviation of the sampling distribution of sample means:

$\sigma_{\bar x}=\cfrac{\sigma}{\sqrt n}=\cfrac{1.58}{\sqrt 2}=1.12.$