To find the standard deviation of the sampling distribution (also called the standard error of the mean) for a sample size $n = 30$ and a population standard deviation $\sigma = 19$, we can use the formula for the standard error:

$\text{Standard Error (SE)} = \frac{\sigma}{\sqrt{n}}$

Where:

• $\sigma = 19$ (population standard deviation),
• $n = 30$ (sample size).

Now, we can compute the standard error.

$SE = \frac{19}{\sqrt{30}} \approx \frac{19}{5.477} \approx 3.47$

Thus, the standard deviation of the sampling distribution is approximately 3.47.

Would you like more details or have any questions?

Here are some related questions:

1. How does increasing the sample size affect the standard error?
2. What is the relationship between sample size and variability in sampling distributions?
3. How would the result change if the population standard deviation was unknown?
4. What is the difference between standard deviation and standard error?
5. Why is it important to use the population standard deviation in this context?

Tip: The larger the sample size, the smaller the standard error, meaning the sample mean is more precise.