SOLUTION: A population consist of five persons those age are 10,20,30,40, and 50. A random sample of size two is to be selected from the population with replacement. 1: list all the possib

Question 1177421: A population consist of five persons those age are 10,20,30,40, and 50.
A random sample of size two is to be selected from the population with replacement.
1: list all the possible sample of two persons age computing the mean of each sample.
2: Obtain the sampling distribution of the mean.
3:Draw a line graph representing distribution and discribe it.
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
**1. List of Possible Samples and Their Means**
Since we're sampling with replacement, the same person can be selected twice. Here's a list of all possible samples and their means:
| Sample | Ages | Mean |
|---|---|---|
| 1 | 10, 10 | 10 |
| 2 | 10, 20 | 15 |
| 3 | 10, 30 | 20 |
| 4 | 10, 40 | 25 |
| 5 | 10, 50 | 30 |
| 6 | 20, 10 | 15 |
| 7 | 20, 20 | 20 |
| 8 | 20, 30 | 25 |
| 9 | 20, 40 | 30 |
| 10 | 20, 50 | 35 |
| 11 | 30, 10 | 20 |
| 12 | 30, 20 | 25 |
| 13 | 30, 30 | 30 |
| 14 | 30, 40 | 35 |
| 15 | 30, 50 | 40 |
| 16 | 40, 10 | 25 |
| 17 | 40, 20 | 30 |
| 18 | 40, 30 | 35 |
| 19 | 40, 40 | 40 |
| 20 | 40, 50 | 45 |
| 21 | 50, 10 | 30 |
| 22 | 50, 20 | 35 |
| 23 | 50, 30 | 40 |
| 24 | 50, 40 | 45 |
| 25 | 50, 50 | 50 |
**2. Sampling Distribution of the Mean**
Now, let's organize the sample means into a frequency distribution:
| Sample Mean | Frequency | Probability |
|---|---|---|
| 10 | 1 | 1/25 |
| 15 | 2 | 2/25 |
| 20 | 3 | 3/25 |
| 25 | 4 | 4/25 |
| 30 | 5 | 5/25 |
| 35 | 4 | 4/25 |
| 40 | 3 | 3/25 |
| 45 | 2 | 2/25 |
| 50 | 1 | 1/25 |
This table represents the sampling distribution of the mean. It shows the probability of obtaining each sample mean when drawing a random sample of size two with replacement from the population.
**3. Line Graph and Description**
Here's a line graph representing the sampling distribution:
[asy]
unitsize(1 cm);
int i;
real prob;
draw((0,0)--(10,0));
draw((0,0)--(0,0.25));
label("Sample Mean", (10,0), E);
label("Probability", (0,0.25), N);
for (i = 1; i <= 9; ++i) {
prob = (i <= 5 ? i : 10 - i)/25;
draw((i,prob)--(i + 1,prob));
dot((i,prob));
}
label("$10$", (1,0), S);
label("$15$", (2,0), S);
label("$20$", (3,0), S);
label("$25$", (4,0), S);
label("$30$", (5,0), S);
label("$35$", (6,0), S);
label("$40$", (7,0), S);
label("$45$", (8,0), S);
label("$50$", (9,0), S);
[/asy]
**Description:**
The line graph shows a symmetric distribution centered around the mean of 30. The distribution is unimodal, with the highest probability at the mean. As you move away from the mean, the probabilities decrease. This shape is somewhat similar to a normal distribution, although it's more discrete and triangular in this case due to the small sample size and the nature of the original population.
**Key Observations:**
* The mean of the sampling distribution is equal to the population mean (30).
* The sampling distribution has less variability than the original population.
* Even though the original population is not normally distributed, the sampling distribution starts to exhibit some characteristics of a normal distribution. This is a hint of the Central Limit Theorem, which states that the sampling distribution of the mean approaches a normal distribution as the sample size increases.
RELATED QUESTIONS
A population consist of five persons those ages are 10, 20,30,40 and 50 years. A random... (answered by CPhill)

A random sample of size 36 is to be selected A from a population that has a mean μ = (answered by ewatrrr)

suppose it is known that 30% of a certain population are immune to some disease. if a... (answered by stanbon,ewatrrr)

A simple random sample of size 3 is to be drawn from the population of 10 students.... (answered by ewatrrr)

A random sample is selected from a normal population with a mean of u=40 and a standard... (answered by jim_thompson5910)

30% of the population of particular country favour the political party BNM. Suppose... (answered by ewatrrr)

explain why a random sample of size 50 is to be preferred to a random sample of size 35... (answered by Boreal)

mu=40, sigma=10. If random samples of size n = 25 are selected from this population, what (answered by stanbon)

Consider a normal population with a mean of 50 and standard deviation of 2. A random... (answered by Boreal)