1.For a standard normal variable Z, which of the following z values satisfies the expression: P(Z > z) = 0.57

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.

Learn how to find the z-value for which P(Z > z) = 0.57 using the standard normal distribution. This problem involves understanding probabilities in the context of normal distributions and using the cumulative distribution function (CDF). Suitable for advanced high school students and anyone studying statistics or probability.

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