For a normally distributed random variable, if standard deviation is 6 and x = 67, p = 0.0038:

What is the mean?

Answer:

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.

Learn how to find the mean of a normally distributed random variable when given the standard deviation, a specific value, and its corresponding probability. This problem involves using the properties of normal distribution and the z-score formula. Suitable for advanced high school students and anyone interested in probability and statistics.

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