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We are tasked with using the **Central Limit Theorem (CLT)** to find the **mean** and the **standard error** of the mean for the sampling distribution.

### Given Information:
- Population mean (\(\mu\)) = 5.1 years
- Population standard deviation (\(\sigma\)) = 1.8 years
- Sample size (\(n\)) = 20

---

### 1. **Mean of the Sampling Distribution**  
According to the **Central Limit Theorem**, the mean of the sampling distribution of the sample mean is equal to the population mean:  
\[
\mu_{\bar{x}} = \mu
\]
Thus:  
\[
\mu_{\bar{x}} = 5.1 \ \text{years}
\]

---

### 2. **Standard Error of the Mean (SE)**  
The standard error of the mean is calculated using the formula:  
\[
\text{SE} = \frac{\sigma}{\sqrt{n}}
\]
Substitute the values:  
\[
\text{SE} = \frac{1.8}{\sqrt{20}}
\]

First, calculate \(\sqrt{20}\):  
\[
\sqrt{20} \approx 4.47
\]

Now compute SE:  
\[
\text{SE} = \frac{1.8}{4.47} \approx 0.403
\]

---

### Final Answers:
1. **Mean of the sampling distribution**: \( 5.1 \ \text{years} \)  
2. **Standard error of the mean**: \( 0.403 \ \text{years} \)

Would you like a further breakdown or explanation of any step?

---

### Related Questions:
1. What happens to the standard error if the sample size increases to 50?
2. How does the Central Limit Theorem apply when the population is not normally distributed?
3. What is the probability that a sample mean falls within 1 standard error of the population mean?
4. How would you interpret the standard error in practical terms?
5. What changes occur in the sampling distribution if the population standard deviation doubles?

**Tip**: As the sample size increases, the standard error decreases, making the sample means more precise estimates of the population mean.

Use the Central Limit Theorem to find the mean and standard error of the mean of the indicated sampling distribution. The amounts of time employees of a telecommunications company have worked for the company are normally distributed with a mean of 5.1 years and a standard deviation of 1.8 years. Random samples of size 20 are drawn from the population and the mean of each sample is determined.

Learn how to use the Central Limit Theorem to calculate the mean and standard error of the sampling distribution for employee work durations. Given a mean of 5.1 years, standard deviation of 1.8 years, and sample size of 20, this problem explores key statistical concepts with detailed explanations.

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