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We are given that \( X \sim N(-46.9, 12.5) \), meaning \( X \) follows a normal distribution with a mean \( \mu = -46.9 \) and a standard deviation \( \sigma = 12.5 \). We need to find the value of \( a \) such that \( P(X < a) = 0.65 \).

### Steps to solve:
1. **Find the z-score corresponding to a probability of 0.65** using the standard normal distribution table or a calculator.
   - For a cumulative probability of 0.65, the corresponding z-score is approximately **0.385**.
   
2. **Convert the z-score to the value of \( a \)** using the formula:
   \[
   z = \frac{a - \mu}{\sigma}
   \]
   Substituting the known values:
   \[
   0.385 = \frac{a - (-46.9)}{12.5}
   \]
   
3. **Solve for \( a \)**:
   \[
   a - (-46.9) = 0.385 \times 12.5
   \]
   \[
   a + 46.9 = 4.8125
   \]
   \[
   a = 4.8125 - 46.9
   \]
   \[
   a = -42.0875
   \]

Thus, the value of \( a \) is approximately **-42.09**.

### Z-scores needed:
The z-score used in this case is **0.385**.

Would you like further details or have any questions?

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Here are some related questions for deeper understanding:
1. What is the relationship between z-scores and percentiles in a normal distribution?
2. How do we use z-scores to standardize values from different normal distributions?
3. What is the cumulative distribution function (CDF) in a normal distribution?
4. How does changing the mean \( \mu \) affect the normal distribution curve?
5. How would the calculation change if \( P(X < a) = 0.75 \) instead of 0.65?

**Tip:** Always check whether you're dealing with a left-tailed or right-tailed probability when interpreting z-scores.

Let X ~ N(-46.9, 12.5). Find a so that P(X < a) = 0.65. List the z-scores needed to calculate the result. If there is more than one z-score, separate the values with a comma.

This problem involves finding the value of 'a' for a normal distribution N(-46.9, 12.5) where P(X < a) = 0.65. The solution includes calculating the z-score and converting it to the corresponding value of 'a'. Suitable for high school students in Grades 11-12.

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