To find the value of $z$ for which $P(Z > z) = 0.57$, we can use the standard normal distribution table (also known as the Z-table) or the cumulative distribution function (CDF) of the standard normal distribution.

Step-by-Step Solution:

1. Understand the problem:

   • $P(Z > z) = 0.57$ means that the probability that the standard normal variable $Z$ is greater than $z$ is 0.57.

2. Translate to cumulative probability:

   • The cumulative distribution function (CDF) gives $P(Z \leq z)$, which is the area to the left of $z$.
   • Since the total probability is 1, we have: $P(Z \leq z) = 1 - P(Z > z) = 1 - 0.57 = 0.43$
   • So, we need to find $z$ such that $P(Z \leq z) = 0.43$.

3. Find the corresponding z-value:

   • Using a standard normal distribution table or a calculator, find the $z$-value that corresponds to a cumulative probability of 0.43.
   • Looking it up, $z \approx -0.18$.

Final Answer:

The value of $z$ that satisfies $P(Z > z) = 0.57$ is approximately $z \approx -0.18$.

Would you like a more detailed explanation, or do you have any questions?

Here are some related questions you might find helpful:

1. What is the standard normal distribution?
2. How do you interpret a Z-score?
3. How do you use the Z-table to find probabilities?
4. What is the cumulative distribution function (CDF)?
5. How would you find the probability $P(Z < z)$ for a given z-value?
6. How do you calculate the probability between two Z-scores?
7. How do you find Z-scores that correspond to specific probabilities without a Z-table?
8. What are some applications of Z-scores in real-life scenarios?

Tip: Remember that for standard normal distribution, negative Z-scores indicate values below the mean, and positive Z-scores indicate values above the mean.