Answer to Question #255014 in Statistics and Probability for Kevin

Question

Lesson Activity: A. A population consists of the numbers 2, 4, 8, 10 and 5. Let us list all the possible samples of size 3 from this population and construct the sampling distribution of the sample mean. B. Complete the statement by filling in the blank. Write your answer on a separate sheet of paper. A ___________ is a frequency distribution using the means computed from all possible random samples of a specific size taken from a population. To get the possible samples use the formula ______, where N is the ________ and n is the ____ size to be taken. The total probability of the sample mean must be equal to ____.

Answer 1

Since the size of the population is $N=5$ sample sizes of $n=3$ are to be drawn from this population then, there are $\binom{N}{n}=\binom{5}{3}=5!/(3!*2!)=10$ possible samples that can be drawn.

The possible samples and their respective sample means are given below.

The sample means are derived using the formula below,

$\bar{x}=\sum(x)/n$ where $x$ are the data values for each sample and $n=3$ for each sample.

$sample$ $\bar{x}$

2,4,8 (2+4+8)/3=4.67

2,4,10 (2+4+10)/3=5.33

2,4,5 (2+4+5)/3=3.67

2,8,10 (2+8+10)/3=6.67

2,8,5 (2+8+5)/3=5

2,10,5 (2+10+5)/3=5.67

4,8,10 (4+8+10)/3=7.33

4,8,5 (4+8+5)/3=5.67

4,10,5 (4+10+5)/3=6.33

8,10,5 (8+10+5)/3=7.67

Since we are drawing at random, each sample will have the same probability of being chosen. Therefore probabilities for these sample means are,

$p(\bar{x}=4.67)=1/10$

$p(\bar{x}=5.33)=1/10$

$p(\bar{x}=3.67)=1/10$

$p(\bar{x}=6.67)=1/10$

$p(\bar{x}=5)=1/10$

$p(\bar{x}=5.67)=1/5$

$p(\bar{x}=7.33)=1/10$

$p(\bar{x}=6.33)=1/10$

$p(\bar{x}=7.67)=1/10$

A $sampling\space distribution$ is a frequency distribution using the means computed from all possible random samples of a specific size taken from a population. To get the possible samples use the formula $\binom{N}{n}=N!/(n!*(N-n)!)$, where $N$ is the $population\space size$ and $n$ is the $sample$ size to be taken. The total probability of the sample mean must be equal to $1$.