A simple random sample of size nequals45 is obtained from a population that is skewed left with muequals42 and sigmaequals9. Does the population need to be normally distributed for the sampling distribution of x overbar to be approximately normally​ distributed? Why? What is the sampling distribution of x overbar​?
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Part 1
Does the population need to be normally distributed for the sampling distribution of x overbar to be approximately normally​ distributed? Why?
A.
Yes. The central limit theorem states that the sampling variability of nonnormal populations will increase as the sample size increases.
B.
No. The central limit theorem states that regardless of the shape of the underlying​ population, the sampling distribution of x overbar becomes approximately normal as the sample​ size, n, increases.
C.
No. The central limit theorem states that only if the shape of the underlying population is normal or uniform does the sampling distribution of x overbar become approximately normal as the sample​ size, n, increases.
D.
Yes. The central limit theorem states that only for underlying populations that are normal is the shape of the sampling distribution of x overbar ​normal, regardless of the sample​ size, n.

To determine whether the population needs to be normally distributed for the sampling distribution of \(\overline{x}\) (the sample mean) to be approximately normal, we can use the Central Limit Theorem (CLT).

**Central Limit Theorem Explanation:**
The Central Limit Theorem states that the sampling distribution of the sample mean \(\overline{x}\) will be approximately normally distributed if the sample size \(n\) is sufficiently large, regardless of the shape of the population distribution. A common rule of thumb is that the sample size should be at least 30 for the CLT to hold well, especially if the population distribution is not normal.

**Application to the Problem:**
- You have a sample size \(n = 45\), which is greater than 30.
- The population distribution is skewed left, which means it is not normal.

**Answer to Part 1:**
The correct answer is **B**. No. The central limit theorem states that regardless of the shape of the underlying population, the sampling distribution of \(\overline{x}\) becomes approximately normal as the sample size, \(n\), increases.

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**Sampling Distribution of \(\overline{x}\):**
The sampling distribution of the sample mean \(\overline{x}\) will have the following properties:
1. **Mean:** \(\mu_{\overline{x}} = \mu = 42\)
2. **Standard Deviation (Standard Error):** \(\sigma_{\overline{x}} = \frac{\sigma}{\sqrt{n}} = \frac{9}{\sqrt{45}}\)

Let's calculate the standard deviation of the sampling distribution:

\[
\sigma_{\overline{x}} = \frac{9}{\sqrt{45}} \approx \frac{9}{6.71} \approx 1.34
\]

**Conclusion:**
- The mean of the sampling distribution of \(\overline{x}\) is 42.
- The standard deviation (standard error) of the sampling distribution of \(\overline{x}\) is approximately 1.34.
- The sampling distribution is approximately normal due to the Central Limit Theorem.

**Do you have any questions, or would you like further details on any part of this explanation?**

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Here are some related questions to expand on this topic:

1. How does the Central Limit Theorem apply to populations with different shapes?
2. Why is a sample size of 30 often considered sufficient for the Central Limit Theorem?
3. What happens to the sampling distribution if the sample size is much smaller than 30?
4. How does the standard error change with larger sample sizes?
5. What is the impact of population skewness on small sample sizes?
6. How can we verify normality in the sampling distribution practically?
7. What are some exceptions or limitations to the Central Limit Theorem?
8. How does increasing the sample size improve the approximation of normality?

**Tip:** When dealing with real-world data, always consider the sample size and the Central Limit Theorem to understand the behavior of sampling distributions effectively.

Explore how the Central Limit Theorem applies to sampling distributions with a skewed left population. Learn why sample size affects the normality of the sampling distribution of \(\overline{x}\) and calculate standard error. Suitable for undergraduate students studying statistics.

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