To find the probability $P(0 < z < 1.59)$ for the standard normal distribution, you can use the standard normal table (often called the Z-table), which provides the cumulative probability from the far left of the distribution up to a given $z$-value.

Steps to solve:

1. Find $P(z < 1.59)$:
   Use the Z-table to find the cumulative probability corresponding to $z = 1.59$.

   • Look up the value for $z = 1.59$ in the standard normal table. The value you find in the table is the cumulative probability up to $z = 1.59$.

   For $z = 1.59$, the cumulative probability is approximately 0.9441.

2. Find $P(z < 0)$:
   Since $z = 0$ is the mean of the standard normal distribution, the cumulative probability at $z = 0$ is exactly 0.5.

3. Calculate $P(0 < z < 1.59)$:
   The probability of $z$ being between 0 and 1.59 is the difference between the cumulative probabilities at $z = 1.59$ and $z = 0$:

   $P(0 < z < 1.59) = P(z < 1.59) - P(z < 0)$ $P(0 < z < 1.59) = 0.9441 - 0.5 = 0.4441$

Final Answer:

$P(0 < z < 1.59) = 0.4441$

Would you like further details or have any questions?

---

Here are five related questions to consider:

1. How do you find the probability for $z$ values greater than 1.59?
2. What is the probability of $z$ being between -1 and 1.59?
3. How do you use the Z-table for negative $z$ values?
4. What does a $z$-score represent in a normal distribution?
5. How would you calculate $P(z > 1.59)$?

Tip: Always remember that the standard normal distribution is symmetrical, and the total area under the curve equals 1.