A random variable is known to be normally distributed with a standard deviation of 0.5. For a an x value of 6.1425, what should be the population mean so that the probability of a value from this distribution exceeding 6.1425 is at most 0.05.
 A. 3.98
 B. 4.47
 C. 6.02
 D. 5.32

To find the population mean \(\mu\) such that the probability of a value exceeding 6.1425 is at most 0.05, we can use the properties of the normal distribution.

### Step-by-step solution:

1. **Interpret the problem:**
   The problem states that the random variable follows a normal distribution with a **standard deviation** \(\sigma = 0.5\). The goal is to find the mean \(\mu\) such that the probability of the random variable \(X\) exceeding \(x = 6.1425\) is at most 0.05.

   Mathematically, we are looking for:
   \[
   P(X > 6.1425) = 0.05
   \]
   which means we are interested in the upper tail of the normal distribution.

2. **Use the Z-score:**
   To find this probability, we first need to convert the value of 6.1425 to a Z-score. The Z-score formula is:
   \[
   Z = \frac{x - \mu}{\sigma}
   \]
   where \(x = 6.1425\), \(\mu\) is the mean, and \(\sigma = 0.5\).

3. **Find the Z-score corresponding to the 95th percentile:**
   Since we want the probability to be at most 0.05, we need to find the Z-score that corresponds to the cumulative probability of 0.95 (because 0.05 is in the upper tail, and the cumulative probability to the left of the Z-value should be 0.95).

   From standard Z-tables or using statistical software, the Z-score corresponding to a cumulative probability of 0.95 is approximately \(Z = 1.645\).

4. **Set up the equation:**
   Using the Z-score formula, we have:
   \[
   1.645 = \frac{6.1425 - \mu}{0.5}
   \]
   
5. **Solve for \(\mu\):**
   Multiply both sides by 0.5:
   \[
   1.645 \times 0.5 = 6.1425 - \mu
   \]
   \[
   0.8225 = 6.1425 - \mu
   \]
   Rearranging:
   \[
   \mu = 6.1425 - 0.8225
   \]
   \[
   \mu = 5.32
   \]

Thus, the population mean \(\mu\) should be **5.32**.

### Final Answer: 
**D. 5.32**

---

### Would you like a detailed explanation of any of these steps? 

Here are 5 related questions to explore:

1. How do we calculate the Z-score for other cumulative probabilities?
2. What is the interpretation of a Z-score in a normal distribution?
3. How can we apply the same method to find the mean for a different probability, say 0.01 or 0.10?
4. How do we solve for the Z-score when given a specific probability threshold?
5. How does the standard deviation affect the Z-score and the corresponding value?

**Tip**: Always remember that the Z-score tells you how many standard deviations a given value is away from the mean!

A random variable is known to be normally distributed with a standard deviation of 0.5. For an x value of 6.1425, what should be the population mean so that the probability of a value from this distribution exceeding 6.1425 is at most 0.05.

Learn how to find the population mean in a normally distributed random variable, ensuring that the probability of exceeding a given x value is at most 0.05. This problem explores the Z-score calculation and properties of the normal distribution, including step-by-step solutions. Suitable for students in Grades 10-12.

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