The problem asks to find the value of $a$ such that:

$P(Z > a) = 0.69$

where $Z \sim N(0,1)$, meaning $Z$ follows a standard normal distribution (with mean 0 and variance 1).

Step-by-Step Solution:

1. Relation to Standard Normal Distribution: The probability $P(Z > a) = 0.69$ can be rewritten using the complementary rule of probabilities as: $P(Z \leq a) = 1 - 0.69 = 0.31$ This means we are looking for the value of $a$ such that the cumulative distribution function (CDF) of the standard normal distribution equals 0.31.

2. Using the Z-table or Inverse CDF: The value of $a$ corresponds to the inverse of the standard normal CDF for a probability of 0.31. Using a Z-table or a calculator that computes the inverse of the CDF for a standard normal distribution, we find that: $a \approx -0.4973$

Thus, the value of $a$ is approximately $-0.4973$.

Would you like more details on how this calculation is performed or have any further questions?

Here are 5 related questions to explore:

1. What is the relationship between the CDF and the PDF of the normal distribution?
2. How can you find probabilities for a normal distribution that is not standard (i.e., with different mean and variance)?
3. What is the difference between $P(Z > a)$ and $P(Z \leq a)$?
4. How do you interpret probabilities in the context of a standard normal distribution?
5. What is the significance of a Z-score in statistics?

Tip: When working with normal distributions, always check if it's standard or not, as non-standard normal distributions require transformations before using Z-tables or inverse CDFs.