To find the z-scores corresponding to the given cumulative probabilities $P(Z \leq z)$, you typically use the standard normal distribution table (z-table) or a calculator. Let's calculate each one:

(a) $z_{0.4236}$

For $P(Z \leq z) = 0.4236$, we need to find the z-value that corresponds to a cumulative probability of 0.4236.

Looking at the z-table, this value is approximately:

• $z = -0.19$

(b) $z_{0.3997}$

For $P(Z \leq z) = 0.3997$, check the z-table for this cumulative probability.

The corresponding z-value is approximately:

• $z = -0.25$

(c) $z_{0.3790}$

For $P(Z \leq z) = 0.3790$, look up the closest cumulative probability in the z-table.

The corresponding z-value is approximately:

• $z = -0.31$

(d) $z_{0.4946}$

For $P(Z \leq z) = 0.4946$, find the cumulative probability.

The corresponding z-value is approximately:

• $z = -0.01$

Would you like more details or have any questions about the z-table process? Here are 5 related questions for deeper exploration:

1. How do you interpret a z-score in statistics?
2. What is the relation between the z-score and cumulative probability?
3. How would you calculate a z-score without a z-table?
4. What is the significance of a negative z-score?
5. How can z-scores be used in hypothesis testing?

Tip: When using a z-table, always verify whether you're reading from a standard normal cumulative table (for $P(Z \leq z)$) or the tail probability table.