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
challen 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: Identify population elements
Population: {3, 5, 7}. Sample size $ n = 2 $, with replacement (so we can repeat elements).
## Step2: List all combinations
- (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 mean for each sample
Mean formula: $ \bar{x} = \frac{x_1 + x_2}{2} $
- (3, 3): $ \frac{3 + 3}{2} = 3 $
- (3, 5): $ \frac{3 + 5}{2} = 4 $
- (3, 7): $ \frac{3 + 7}{2} = 5 $
- (5, 3): $ \frac{5 + 3}{2} = 4 $
- (5, 5): $ \frac{5 + 5}{2} = 5 $
- (5, 7): $ \frac{5 + 7}{2} = 6 $
- (7, 3): $ \frac{7 + 3}{2} = 5 $
- (7, 5): $ \frac{7 + 5}{2} = 6 $
- (7, 7): $ \frac{7 + 7}{2} = 7 $
## Step2: Create sampling distribution table (frequency of each mean)
| 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 variance formula for sampling distribution
$ \sigma_{\bar{x}}^2 = \frac{\sum f(\bar{x} - \mu_{\bar{x}})^2}{N} $ (where $ N $ is total number of samples, $ \mu_{\bar{x}} $ is mean of sample means)
First, find $ \mu_{\bar{x}} $:
$ \mu_{\bar{x}} = \frac{\sum f\bar{x}}{\sum f} = \frac{1(3) + 2(4) + 3(5) + 2(6) + 1(7)}{9} = \frac{3 + 8 + 15 + 12 + 7}{9} = \frac{45}{9} = 5 $
## Step2: Calculate $ (\bar{x} - \mu_{\bar{x}})^2 $ for each $ \bar{x} $
- For $ \bar{x} = 3 $: $ (3 - 5)^2 = 4 $, $ f = 1 $, $ f(\bar{x} - \mu_{\bar{x}})^2 = 1 \times 4 = 4 $
- For $ \bar{x} = 4 $: $ (4 - 5)^2 = 1 $, $ f = 2 $, $ f(\bar{x} - \mu_{\bar{x}})^2 = 2 \times 1 = 2 $
- For $ \bar{x} = 5 $: $ (5 - 5)^2 = 0 $, $ f = 3 $, $ f(\bar{x} - \mu_{\bar{x}})^2 = 3 \times 0 = 0 $
- For $ \bar{x} = 6 $: $ (6 - 5)^2 = 1 $, $ f = 2 $, $ f(\bar{x} - \mu_{\bar{x}})^2 = 2 \times 1 = 2 $
- For $ \bar{x} = 7 $: $ (7 - 5)^2 = 4 $, $ f = 1 $, $ f(\bar{x} - \mu_{\bar{x}})^2 = 1 \times 4 = 4 $
## Step3: Sum the squared terms
$ \sum f(\bar{x} - \mu_{\bar{x}})^2 = 4 + 2 + 0 + 2 + 4 = 12 $
## Step4: Compute variance
$ \sigma_{\bar{x}}^2 = \frac{12}{9} = \frac{4}{3} \approx 1.333 $
# Answer:
$ \frac{4}{3} $ (or approximately 1.333)

### 4. Find the standard deviation of the sample means.
# Explanation:
## Step1: Recall standard deviation is square root of variance
$ \sigma_{\bar{x}} = \sqrt{\sigma_{\bar{x}}^2} $
From part 3, $ \sigma_{\bar{x}}^2 = \frac{4}{3} $
## Step2: Take square root
$ \sigma_{\bar{x}} = \sqrt{\frac{4}{3}} = \frac{2}{\sqrt{3}} \approx 1.1547 $
# Answer:
$ \frac{2}{\sqrt{3}} $ (or approximately 1.1547)

### 5. Find the population mean.
# Explanation:
## Step1: Population mean formula
$ \mu = \frac{\sum x}{N} $ (where $ N $ is population size)
Population: {3, 5, 7}, so $ \sum x = 3 + 5 + 7 = 15 $, $ N = 3 $
## Step2: Calculate mean
$ \mu = \frac{15}{3} = 5 $
# Answer:
5

### 6. Find the variance of the population.
# Explanation:
## Step1: Population variance formula
$ \sigma^2 = \frac{\sum (x - \mu)^2}{N} $
From part 5, $ \mu = 5 $
## Step2: Calculate $ (x - \mu)^2 $ for each $ x $
- $ x = 3 $: $ (3 - 5)^2 = 4 $
- $ x = 5 $: $ (5 - 5)^2 = 0 $
- $ x = 7 $: $ (7 - 5)^2 = 4 $
## Step3: Sum and divide by $ N $
$ \sum (x - \mu)^2 = 4 + 0 + 4 = 8 $, $ \sigma^2 = \frac{8}{3} \approx 2.6667 $
# Answer:
$ \frac{8}{3} $ (or approximately 2.6667)

### 7. Find the standard deviation of the population.
# Explanation:
## Step1: Standard deviation is square root of variance
From part 6, $ \sigma^2 = \frac{8}{3} $
## Step2: Take square root
$ \sigma = \sqrt{\frac{8}{3}} = \frac{2\sqrt{6}}{3} \approx 1.63299 $
# Answer:
$ \frac{2\sqrt{6}}{3} $ (or approximately 1.633)

### 8. Find the standard error of the mean.
# Explanation:
## Step1: Standard error formula (with replacement)
$ SE = \frac{\sigma}{\sqrt{n}} $ (where $ \sigma $ is population standard deviation, $ n $ is sample size)
From part 7, $ \sigma = \frac{2\sqrt{6}}{3} $, $ n = 2 $
## Step2: Substitute values
$ SE = \frac{\frac{2\sqrt{6}}{3}}{\sqrt{2}} = \frac{2\sqrt{6}}{3\sqrt{2}} = \frac{2\sqrt{3}}{3} \approx 1.1547 $
(Alternatively, from part 4, $ SE = \sigma_{\bar{x}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3} $, 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: Identify population elements

Step2: List all combinations

Step1: Calculate mean for each sample

Step2: Create sampling distribution table (frequency of each mean)

Step1: Recall variance formula for sampling distribution

Step2: Calculate \( (\bar{x} - \mu_{\bar{x}})^2 \) for each \( \bar{x} \)

Step3: Sum the squared terms

Step4: Compute variance

Answer:

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

Sample Mean (\( \bar{x} \))	Frequency (\( f \))
4	2
5	3
6	2
7	1