What is the sampling distribution of 

x overbarx​? 

Select the correct choice below and fill in the answer boxes within your choice.

​(Type integers or decimals rounded to three decimal places as​ needed.)

A.

The shape of the sampling distribution of 

x overbarx 

is unknown with 

mu Subscript x overbarμxequals=enter your response here 

and 

sigma Subscript x overbarσxequals=enter your response here.

B.

The sampling distribution of 

x overbarx 

is approximately normal with 

mu Subscript x overbarμxequals=enter your response here 

and 

sigma Subscript x overbarσxequals=enter your response here.

C.

The sampling distribution of 

x overbarx 

is skewed left with 

mu Subscript x overbarμxequals=enter your response here 

and 

sigma Subscript x overbarσxequals=enter your response here.

D.

The sampling distribution of 

x overbarx 

is uniform with 

mu Subscript x overbarμxequals=enter your response here 

and 

sigma Subscript x overbarσxequals=enter your response here.

To determine the characteristics of the sampling distribution of \(\overline{x}\), we need to consider the Central Limit Theorem and the given parameters:

- The population mean (\(\mu\)) is 42.
- The population standard deviation (\(\sigma\)) is 9.
- The sample size (\(n\)) is 45.

**Using the Central Limit Theorem:**

1. **Shape:** The Central Limit Theorem tells us that for a sufficiently large sample size (typically \(n \geq 30\)), the sampling distribution of the sample mean \(\overline{x}\) will be approximately normal, regardless of the population distribution's shape.

2. **Mean (\(\mu_{\overline{x}}\)):** The mean of the sampling distribution of \(\overline{x}\) is the same as the population mean:
   \[
   \mu_{\overline{x}} = \mu = 42
   \]

3. **Standard Deviation (Standard Error, \(\sigma_{\overline{x}}\)):** The standard deviation of the sampling distribution, also known as the standard error, is given by:
   \[
   \sigma_{\overline{x}} = \frac{\sigma}{\sqrt{n}} = \frac{9}{\sqrt{45}}
   \]
   Calculating this gives:
   \[
   \sigma_{\overline{x}} = \frac{9}{\sqrt{45}} = \frac{9}{6.708} \approx 1.343
   \]

**Conclusion:**
The correct choice is **B**:

The sampling distribution of \(\overline{x}\) is approximately normal with:
- \(\mu_{\overline{x}} = 42\)
- \(\sigma_{\overline{x}} = 1.343\)

Therefore, the sampling distribution is approximately normal with the specified mean and standard deviation.

**Do you have any questions or need further explanation on any of these points?**

---

Here are some related questions to explore further:

1. What factors affect the accuracy of the Central Limit Theorem's approximation?
2. How would the sampling distribution change if the sample size were smaller, such as \(n = 15\)?
3. Can the Central Limit Theorem be applied to median or other statistics besides the mean?
4. What are some practical examples of using the Central Limit Theorem in data analysis?
5. How do we interpret the standard error in terms of sampling variability?
6. Why is it important to know the shape of the sampling distribution?
7. How does the sampling distribution relate to confidence intervals?
8. What are some misconceptions about the Central Limit Theorem?

**Tip:** Remember that the Central Limit Theorem is a powerful tool in statistics, allowing us to make inferences about population parameters based on sample statistics, even when the population distribution is unknown or non-normal.

Explore the characteristics of the sampling distribution of sample means, focusing on the Central Limit Theorem. Learn how to determine the shape, mean, and standard deviation of the sampling distribution based on given population parameters. Suitable for undergraduate students studying statistics.

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