For a population that is normally distributed with a mean $\mu = 82$ and a standard deviation $\sigma = 12$, we can describe the characteristics of the sampling distribution for the sample mean $\overline{X}$ based on a random sample of size $n = 16$ using the following properties of the sampling distribution:

1. Mean of the Sampling Distribution:

The mean of the sampling distribution of the sample mean $\overline{X}$ is the same as the population mean: $\mu_{\overline{X}} = \mu = 82$

2. Standard Deviation of the Sampling Distribution (Standard Error):

The standard deviation of the sampling distribution (known as the standard error) is given by: $\sigma_{\overline{X}} = \frac{\sigma}{\sqrt{n}}$ Substitute $\sigma = 12$ and $n = 16$: $\sigma_{\overline{X}} = \frac{12}{\sqrt{16}} = \frac{12}{4} = 3$

3. Shape of the Sampling Distribution:

Since the population distribution is normal, the sampling distribution of the sample mean $\overline{X}$ will also be normal, even for smaller sample sizes. This is a property of the Central Limit Theorem (CLT).

Summary:

• Mean of $\overline{X}$ = 82
• Standard Error (Standard Deviation of $\overline{X}$) = 3
• Shape = Normal

These characteristics hold true for a random sample of size 16, as long as the population is normal, or the sample size is sufficiently large for the Central Limit Theorem to apply.

Would you like further explanation or details on how this relates to other concepts, such as confidence intervals or hypothesis testing?

Related Questions:

1. How does sample size affect the standard error?
2. What would happen to the sampling distribution if the sample size were increased to 25?
3. How do you use the standard error in hypothesis testing?
4. Can the sampling distribution be used to estimate population parameters?
5. What is the Central Limit Theorem and how does it apply here?

Tip: When dealing with normal distributions and sample means, always ensure that you adjust the standard deviation for the sample size. Smaller sample sizes result in a larger standard error, making estimates less precise.