be computed when population variance is known using ( se = \frac{sigma}{sqrt{n}} ) (for samples with replacement) and ( se = \frac{sigma}{sqrt{n}} cdot sqrt{\frac{n - n}{n - 1}} ) (for samples without replacement) where ( sigma ) is the population standard deviation. example on the case above, three balls numbered 1, 2, and 3. two samples are drawn. the standard deviation of the population is ( sigma = sqrt{\frac{2}{3}} ). let us solve for the se. ( se = \frac{sigma}{sqrt{n}} = \frac{sqrt{\frac{2}{3}}}{sqrt{2}} = sqrt{\frac{2}{6}} = 0.577 ) 3. the shape is approximately normal, provided the sample size is large enough, and regardless of the shape of parent distribution. whats more task 5 challenge yourself! a population consists of the numbers 3, 5, 7. 1. enumerate all possible samples of size 2 with replacement. 2. construct a sampling distribution table and compute for the mean of each sample. 3. find the variance of the sample means. 4. find the standard deviation of the sample means. 5. find the population mean. 6. find the variance of the population. 7. find the standard deviation of the population. 8. find the standard error of the mean.

Question

Accepted Answer

### 1. Enumerate all possible samples of size 2 with replacement.
# Explanation:
## Step1: Understand with replacement sampling
When sampling with replacement, after selecting a number, we put it back, so we can select the same number again. The population is {3, 5, 7}, and we need to draw samples of size 2.
The possible samples are: (3, 3), (3, 5), (3, 7), (5, 3), (5, 5), (5, 7), (7, 3), (7, 5), (7, 7)
# Answer:
(3, 3), (3, 5), (3, 7), (5, 3), (5, 5), (5, 7), (7, 3), (7, 5), (7, 7)

### 2. Construct a sampling distribution table and compute for the mean of each sample.
# Explanation:
## Step1: Calculate the mean for each sample
- For (3, 3): $\bar{x}=\frac{3 + 3}{2}=3$
- For (3, 5): $\bar{x}=\frac{3 + 5}{2}=4$
- For (3, 7): $\bar{x}=\frac{3 + 7}{2}=5$
- For (5, 3): $\bar{x}=\frac{5 + 3}{2}=4$
- For (5, 5): $\bar{x}=\frac{5 + 5}{2}=5$
- For (5, 7): $\bar{x}=\frac{5 + 7}{2}=6$
- For (7, 3): $\bar{x}=\frac{7 + 3}{2}=5$
- For (7, 5): $\bar{x}=\frac{7 + 5}{2}=6$
- For (7, 7): $\bar{x}=\frac{7 + 7}{2}=7$

## Step2: Construct the sampling distribution table
We list the sample means and their frequencies.
| Sample Mean ($\bar{x}$) | Frequency ($f$) |
|-------------------------|-----------------|
| 3                       | 1               |
| 4                       | 2               |
| 5                       | 3               |
| 6                       | 2               |
| 7                       | 1               |

# Answer:
| Sample Mean ($\bar{x}$) | Frequency ($f$) |
|-------------------------|-----------------|
| 3                       | 1               |
| 4                       | 2               |
| 5                       | 3               |
| 6                       | 2               |
| 7                       | 1               |

### 3. Find the variance of the sample means.
# Explanation:
## Step1: Recall the formula for variance of a discrete distribution
The formula for the variance of a discrete random variable (here, the sample mean $\bar{x}$) is $s^2=\frac{\sum f(\bar{x}-\mu_{\bar{x}})^2}{N}$, where $\mu_{\bar{x}}$ is the mean of the sample means, $f$ is the frequency, and $N$ is the total number of samples. First, we find $\mu_{\bar{x}}$.

## Step2: Calculate the mean of the sample means ($\mu_{\bar{x}}$)
$\mu_{\bar{x}}=\frac{\sum f\bar{x}}{\sum f}$
$\sum f\bar{x}=(3	imes1)+(4	imes2)+(5	imes3)+(6	imes2)+(7	imes1)=3 + 8 + 15 + 12 + 7 = 45$
$\sum f=1 + 2 + 3 + 2 + 1 = 9$
So, $\mu_{\bar{x}}=\frac{45}{9}=5$

## Step3: Calculate the sum of $f(\bar{x}-\mu_{\bar{x}})^2$
- For $\bar{x}=3$: $f(\bar{x}-\mu_{\bar{x}})^2=1	imes(3 - 5)^2=1	imes4 = 4$
- For $\bar{x}=4$: $f(\bar{x}-\mu_{\bar{x}})^2=2	imes(4 - 5)^2=2	imes1 = 2$
- For $\bar{x}=5$: $f(\bar{x}-\mu_{\bar{x}})^2=3	imes(5 - 5)^2=3	imes0 = 0$
- For $\bar{x}=6$: $f(\bar{x}-\mu_{\bar{x}})^2=2	imes(6 - 5)^2=2	imes1 = 2$
- For $\bar{x}=7$: $f(\bar{x}-\mu_{\bar{x}})^2=1	imes(7 - 5)^2=1	imes4 = 4$

$\sum f(\bar{x}-\mu_{\bar{x}})^2=4 + 2 + 0 + 2 + 4 = 12$

## Step4: Calculate the variance
$s^2=\frac{12}{9}=\frac{4}{3}\approx1.333$
# Answer:
$\frac{4}{3}$ (or approximately 1.333)

### 4. Find the standard deviation of the sample means.
# Explanation:
## Step1: Recall the formula for standard deviation
The standard deviation of the sample means is the square root of the variance of the sample means. So, $s=\sqrt{s^2}$, where $s^2$ is the variance we found in part 3.

## Step2: Calculate the standard deviation
We know $s^2=\frac{4}{3}$, so $s=\sqrt{\frac{4}{3}}=\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}\approx1.1547$
# Answer:
$\frac{2\sqrt{3}}{3}$ (or approximately 1.1547)

### 5. Find the population mean.
# Explanation:
## Step1: Recall the formula for population mean
The population mean $\mu$ is given by $\mu=\frac{\sum x}{N}$, where $x$ are the population values and $N$ is the population size.

## Step2: Calculate the population mean
The population is {3, 5, 7}, so $\sum x=3 + 5 + 7 = 15$ and $N = 3$.
$\mu=\frac{15}{3}=5$
# Answer:
5

### 6. Find the variance of the population.
# Explanation:
## Step1: Recall the formula for population variance
The population variance $\sigma^2=\frac{\sum (x-\mu)^2}{N}$, where $\mu$ is the population mean, $x$ are the population values, and $N$ is the population size.

## Step2: Calculate $(x-\mu)^2$ for each population value
- For $x = 3$: $(3 - 5)^2=(-2)^2 = 4$
- For $x = 5$: $(5 - 5)^2=0^2 = 0$
- For $x = 7$: $(7 - 5)^2=2^2 = 4$

## Step3: Calculate $\sum (x-\mu)^2$
$\sum (x-\mu)^2=4 + 0 + 4 = 8$

## Step4: Calculate the population variance
$\sigma^2=\frac{8}{3}\approx2.6667$
# Answer:
$\frac{8}{3}$ (or approximately 2.6667)

### 7. Find the standard deviation of the population.
# Explanation:
## Step1: Recall the formula for population standard deviation
The population standard deviation $\sigma$ is the square root of the population variance $\sigma^2$.

## Step2: Calculate the population standard deviation
We know $\sigma^2=\frac{8}{3}$, so $\sigma=\sqrt{\frac{8}{3}}=\frac{2\sqrt{6}}{3}\approx1.63299$
# Answer:
$\frac{2\sqrt{6}}{3}$ (or approximately 1.6330)

### 8. Find the standard error of the mean.
# Explanation:
## Step1: Recall the formula for standard error (with replacement)
The standard error of the mean (SE) when sampling with replacement is $SE=\frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ is the sample size.

## Step2: Identify the values of $\sigma$ and $n$
From part 7, $\sigma=\frac{2\sqrt{6}}{3}$, and the sample size $n = 2$.

## Step3: Calculate the standard error
$SE=\frac{\frac{2\sqrt{6}}{3}}{\sqrt{2}}=\frac{2\sqrt{6}}{3\sqrt{2}}=\frac{2\sqrt{3}}{3}\approx1.1547$ (We can also use the relationship that for sampling with replacement, $SE=\sqrt{	ext{variance of sample means}}$, and from part 3, the variance of sample means is $\frac{4}{3}$, so $SE=\sqrt{\frac{4}{3}}=\frac{2\sqrt{3}}{3}$, which is the same result.)
# Answer:
$\frac{2\sqrt{3}}{3}$ (or approximately 1.1547)

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QUESTION IMAGE

Question

Explanation:

1. Enumerate all possible samples of size 2 with replacement.

Step1: Understand with replacement sampling

Step1: Calculate the mean for each sample

Step2: Construct the sampling distribution table

Step1: Recall the formula for variance of a discrete distribution

Step2: Calculate the mean of the sample means ($\mu_{\bar{x}}$)

Step3: Calculate the sum of $f(\bar{x}-\mu_{\bar{x}})^2$

Step4: Calculate the variance

Answer:

2. Construct a sampling distribution table and compute for the mean of each sample.