To find the mean of a normally distributed random variable when the standard deviation ($\sigma$) is given, along with a specific value of $x$ and its corresponding probability $p$, you can use the properties of the standard normal distribution.

Steps:

1. Identify the z-score corresponding to the given probability.
   $p = 0.0038$ corresponds to a very low tail probability in the standard normal distribution table.

2. Use the z-score formula:
   $z = \frac{x - \mu}{\sigma}$ Here, $\mu$ is the mean, $\sigma$ is the standard deviation, and $x$ is the value for which the probability is given.

3. Find the z-score:
   From the z-table, a probability $p = 0.0038$ corresponds to a z-score of approximately $-2.67$.

4. Substitute the values into the z-score formula:
   $-2.67 = \frac{67 - \mu}{6}$

5. Solve for $\mu$ (the mean):
   $\mu = 67 - (2.67 \times 6)$ $\mu = 67 - 16.02$ $\mu = 50.98$

Final Answer:

The mean $\mu$ is approximately 50.98.

Would you like more details or have any questions? Here are some related questions:

1. How is the z-score derived for different probabilities?
2. What is the significance of the mean in a normal distribution?
3. How do you interpret the standard deviation in the context of normal distribution?
4. Can you explain why the z-score for $p = 0.0038$ is negative?
5. What would happen if the standard deviation were larger or smaller?
6. How can we find the probability for a different value of $x$ using the mean and standard deviation?
7. What are the applications of normal distribution in real-world scenarios?
8. How do you use the standard normal table to find probabilities?

Tip: Always ensure to check the tail of the distribution when looking up probabilities in the z-table.