https://file.aiquickdraw.com/mathbot/file/1734197542666brw1jzlf.jpg

We are tasked with using the **Central Limit Theorem (CLT)** to determine the **mean** and **standard error** of the mean for a sampling distribution. Let’s analyze the given data and solution step-by-step:

---

### **Given Information:**
1. **Population Mean (μ):** \( 5.5 \) years  
2. **Population Standard Deviation (σ):** \( 1.9 \) years  
3. **Sample Size (n):** \( 16 \)

---

### **Solution:**

#### 1. **Mean of the Sampling Distribution (μ\_x̄):**
The Central Limit Theorem tells us that the mean of the sampling distribution of the sample mean is equal to the population mean.

\[
\mu_{\bar{x}} = \mu = 5.5 \, \text{years}
\]

---

#### 2. **Standard Error of the Mean (SE):**
The standard error of the mean is calculated using the formula:

\[
\text{SE} = \frac{\sigma}{\sqrt{n}}
\]

Here:
- \( \sigma = 1.9 \) years  
- \( n = 16 \)  

Substitute the values:

\[
\text{SE} = \frac{1.9}{\sqrt{16}} = \frac{1.9}{4} = 0.475 \, \text{years}
\]

---

### **Final Answers:**
1. **Mean of the Sampling Distribution:** \( 5.5 \, \text{years} \)  
2. **Standard Error (SE):** \( 0.475 \, \text{years} \)

---

Let me know if you need further clarification or a deeper explanation!

---

### **Related Questions to Expand Your Understanding:**
1. How does increasing the sample size affect the standard error of the mean?  
2. What happens to the sampling distribution if the population is not normally distributed?  
3. Can the Central Limit Theorem be applied for very small sample sizes?  
4. How do you interpret the standard error of the mean in real-world problems?  
5. Why is the sampling distribution of the mean important in statistics?  

---

### **Tip:**
The Central Limit Theorem is powerful because it allows us to use the normal distribution to approximate sample means even when the population distribution is not normal, provided the sample size is large enough.

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 5.5 years and a standard deviation of 1.9 years. Random samples of size 16 are drawn from the population and the mean of each sample is determined.

Learn how to calculate the mean and standard error of the sampling distribution using the Central Limit Theorem. Given a mean of 5.5 years, a standard deviation of 1.9 years, and a sample size of 16, determine the standard error step-by-step.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation