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?

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.