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

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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!