This is a short answer question with two parts.  Type your answer in the space below​ (2 to 3​ sentences). ​ NOTE:  Your answers will be graded​ manually, and the final grade will be entered into the gradebook.
Suppose a simple random sample of size nequals64 is obtained from a population that is skewed right with mu equals 76 and sigma equals 24.
​(a) Describe the sampling distribution of x overbar.  Be sure to describe the​ shape, center​ (mean of the sampling​ distribution), and spread​ (standard deviation of the sampling​ distribution).
​(b) How does the sampling distribution of x overbar compare to the population​ distribution? 

### Answer:  

#### (a) Description of the Sampling Distribution of \( \bar{x} \):  
The sampling distribution of \( \bar{x} \) is approximately normal due to the Central Limit Theorem (since \( n = 64 \geq 30 \)). The mean of the sampling distribution (\( \mu_{\bar{x}} \)) is the same as the population mean, \( \mu_{\bar{x}} = \mu = 76 \). The standard deviation of the sampling distribution (\( \sigma_{\bar{x}} \)) is given by \( \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{24}{\sqrt{64}} = 3 \).  

#### (b) Comparison to the Population Distribution:  
The sampling distribution of \( \bar{x} \) is much less skewed than the population distribution, tending towards normality regardless of the population's shape. While the population distribution is skewed right, the sampling distribution of \( \bar{x} \) is symmetric and normal due to the Central Limit Theorem. Additionally, the variability (standard deviation) of the sampling distribution is smaller than the population's variability.

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Would you like further clarification or details?  

Here are 5 related questions to expand your understanding:  
1. Why does the Central Limit Theorem apply in this scenario?  
2. How would the standard deviation of the sampling distribution change if \( n \) increased?  
3. What happens to the shape of the sampling distribution if \( n \) were smaller (e.g., \( n < 30 \))?  
4. How does skewness in the population affect the sampling distribution for small sample sizes?  
5. Can the mean of the sampling distribution ever differ from the population mean? Why or why not?  

**Tip:** Always remember that the Central Limit Theorem ensures normality for large sample sizes, even if the population is not normal.

This is a short answer question with two parts. Type your answer in the space below​ (2 to 3​ sentences). ​ NOTE:  Your answers will be graded​ manually, and the final grade will be entered into the gradebook.
Suppose a simple random sample of size nequals64 is obtained from a population that is skewed right with mu equals 76 and sigma equals 24.
​(a) Describe the sampling distribution of x overbar. Be sure to describe the​ shape, center​ (mean of the sampling​ distribution), and spread​ (standard deviation of the sampling​ distribution).
​(b) How does the sampling distribution of x overbar compare to the population​ distribution?

Understand the sampling distribution of x̄ for a skewed population with μ=76 and σ=24. Learn how the Central Limit Theorem shapes the distribution with a sample size of n=64, and compare the sampling and population distributions. Suitable for Grades 10-12.

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