The question asks to find $a$ such that $P(Z > a) = 0.24$, where $Z \sim N(0, 1)$ is a standard normal distribution.

This is essentially asking for the value of $a$ such that the probability that $Z$ (a standard normal random variable) is greater than $a$ is 0.24. In terms of the cumulative distribution function (CDF) $\Phi(a)$, this can be written as:

$P(Z > a) = 1 - \Phi(a) = 0.24$

Therefore, $\Phi(a) = 1 - 0.24 = 0.76$. Now, we need to find the value of $a$ such that $\Phi(a) = 0.76$, which corresponds to finding the inverse of the standard normal distribution for a probability of 0.76.

Let me calculate that for you.The value of $a$ such that $P(Z > a) = 0.24$ is approximately $a = 0.71$.

Would you like a detailed explanation of the steps, or do you have any questions?

Here are 5 related questions to expand your understanding:

1. What is the significance of the cumulative distribution function (CDF) in statistics?
2. How do you calculate probabilities for non-standard normal distributions?
3. What is the relationship between percentiles and standard scores (z-scores)?
4. How would the calculation change if the probability given was $P(Z < a)$?
5. How does the area under the standard normal curve help in understanding probabilities?

Tip: When working with standard normal distributions, remember that the total area under the curve is 1, and the curve is symmetric about the mean, $Z = 0$.