SOLUTION: A population consist of five persons those ages are 10, 20,30,40 and 50 years. A random sample of size 2 is to be selected from the population with replacement.
(i). List all the
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Question 1177195: A population consist of five persons those ages are 10, 20,30,40 and 50 years. A random sample of size 2 is to be selected from the population with replacement.
(i). List all the possible samples of two persons ages computing the mean, ý of each sample.
(ii). Obtain the sampling distribution of ý.
(ii). Draw a line graph representing the above distribution and describe it.
Answer by CPhill(1987) (Show Source): You can put this solution on YOUR website!
Absolutely, let's break this down step-by-step.
**Population:**
* Ages: 10, 20, 30, 40, 50
* Population size (N): 5
**Sample:**
* Sample size (n): 2
* Sampling method: with replacement
**(i) List all possible samples and their means:**
Since we are sampling with replacement, we can have pairs like (10, 10), (10, 20), etc.
| Sample (Age1, Age2) | Mean (x̄) |
|---|---|
| (10, 10) | 10 |
| (10, 20) | 15 |
| (10, 30) | 20 |
| (10, 40) | 25 |
| (10, 50) | 30 |
| (20, 10) | 15 |
| (20, 20) | 20 |
| (20, 30) | 25 |
| (20, 40) | 30 |
| (20, 50) | 35 |
| (30, 10) | 20 |
| (30, 20) | 25 |
| (30, 30) | 30 |
| (30, 40) | 35 |
| (30, 50) | 40 |
| (40, 10) | 25 |
| (40, 20) | 30 |
| (40, 30) | 35 |
| (40, 40) | 40 |
| (40, 50) | 45 |
| (50, 10) | 30 |
| (50, 20) | 35 |
| (50, 30) | 40 |
| (50, 40) | 45 |
| (50, 50) | 50 |
**(ii) Obtain the sampling distribution of x̄:**
We need to count the frequency of each mean and calculate the probability.
| Mean (x̄) | Frequency | Probability |
|---|---|---|
| 10 | 1 | 1/25 = 0.04 |
| 15 | 2 | 2/25 = 0.08 |
| 20 | 3 | 3/25 = 0.12 |
| 25 | 4 | 4/25 = 0.16 |
| 30 | 5 | 5/25 = 0.20 |
| 35 | 4 | 4/25 = 0.16 |
| 40 | 3 | 3/25 = 0.12 |
| 45 | 2 | 2/25 = 0.08 |
| 50 | 1 | 1/25 = 0.04 |
**(iii) Draw a line graph and describe it:**
```
5 | *
4 | * *
3 | * *
2 | * *
1 | * *
-------------------------
10 15 20 25 30 35 40 45 50 (Mean)
```
**Description:**
* **Symmetrical Distribution:** The line graph is symmetrical around the mean of 30.
* **Bell-Shaped Tendency:** Although it's a discrete distribution, it shows a tendency towards a bell-shaped curve, which is characteristic of sampling distributions of the mean.
* **Center at Population Mean:** The center of the distribution is at 30, which is also the mean of the population (10+20+30+40+50)/5 = 30.
* **Probability:** The probability is highest at the population mean and decreases as you move away from it.
* **Discrete:** Since the sample space is finite, the graph is discrete.
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