A simple random sample of size 𝑛=10 is obtained from a population that is normally distributed with a 𝜇=30 and 𝜎=8.  What is the sampling distribution of 𝑥 ̅?



To determine the sampling distribution of the sample mean (\(\bar{x}\)) for a sample size \(n = 10\) from a normally distributed population, we use the following properties:

### Given Information:
- **Population mean (\(\mu\))** = 30
- **Population standard deviation (\(\sigma\))** = 8
- **Sample size (\(n\))** = 10

### Sampling Distribution of \(\bar{x}\):
1. **Mean of the sampling distribution** (\(\mu_{\bar{x}}\)):
   \[
   \mu_{\bar{x}} = \mu = 30
   \]

2. **Standard deviation of the sampling distribution** (also known as the standard error, \(\sigma_{\bar{x}}\)):
   \[
   \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{8}{\sqrt{10}} \approx 2.53
   \]

3. **Shape of the sampling distribution**:
   Since the population is normally distributed, the sampling distribution of the sample mean \(\bar{x}\) will also be normally distributed, regardless of sample size.

### Conclusion:
The sampling distribution of \(\bar{x}\) is **normally distributed** with:
- Mean, \(\mu_{\bar{x}} = 30\)
- Standard deviation, \(\sigma_{\bar{x}} \approx 2.53\)

So, we can write:
\[
\bar{x} \sim N(30, 2.53)
\]

Would you like further details on this topic?

---

1. How would the sampling distribution change with a larger sample size?
2. How does the Central Limit Theorem affect sampling distributions from non-normal populations?
3. What happens to the standard error if the population standard deviation increases?
4. How would you calculate probabilities for sample means from this distribution?
5. Why is the shape of the sampling distribution normal in this case?

**Tip:** In general, as the sample size \(n\) increases, the standard error of the mean decreases, making the sampling distribution more concentrated around the population mean.

A simple random sample of size n=10 is obtained from a population that is normally distributed with a μ=30 and σ=8. What is the sampling distribution of x̅?

Explore the sampling distribution of the sample mean x̅ from a normally distributed population with mean μ=30 and standard deviation σ=8, using a sample size of n=10. This guide provides key statistical formulas and explains why the distribution of x̅ remains normal. Perfect for introductory statistics learners.

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