We are given that $X \sim N(-42.7, 12)$, meaning that $X$ follows a normal distribution with a mean of $\mu = -42.7$ and a standard deviation of $\sigma = 12$. We need to find $a$ such that $P(X > a) = 0.69$.

Steps to solve:

1. Find the corresponding z-score:
   We first need to find the z-score associated with a probability of 0.69 in the upper tail (since $P(X > a) = 0.69$).

   Using the fact that $P(X > a) = 0.69$, we know that $P(X \leq a) = 1 - 0.69 = 0.31$. Now, we need the z-score that corresponds to a cumulative probability of 0.31.

2. Use the standard normal distribution table (z-table):
   From the z-table, we find that the z-score corresponding to a cumulative probability of 0.31 is approximately $z = -0.50$.

3. Convert the z-score to the value of $a$ using the formula for the z-score: $z = \frac{a - \mu}{\sigma}$ Substituting the known values: $-0.50 = \frac{a - (-42.7)}{12}$ Solving for $a$: $-0.50 = \frac{a + 42.7}{12}$ $a + 42.7 = -0.50 \times 12 = -6$ $a = -6 - 42.7 = -48.7$

Thus, the value of $a$ is $-48.7$.

Z-score:

The z-score needed is $-0.50$.

Would you like more details or have any further questions? Here are some related questions for further practice:

1. What is the probability that $X$ is less than a given value $a$?
2. How do we compute the z-score for a value in a normal distribution?
3. What is the cumulative probability associated with $z = 1.0$?
4. How would you calculate $a$ if $P(X < a) = 0.85$?
5. Can the z-score formula be used for non-normal distributions?

Tip: Always check the direction of the probability (greater or less than) to use the correct side of the z-table.