Find the probability of z occurring in the indicated region of the standard normal distribution.
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0
1.59
z

A normal curve is over a horizontal axis and is centered on 0. Vertical line segments extend from the horizontal axis to the curve at 0 and 1.59, where 1.59 is to the right of 0. The area under the curve between 0 and 1.59 is shaded.
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Part 1
​P(0less thanzless than1.59​)equals

enter your response here
​(Round to four decimal places as​ needed.)

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?

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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.

Learn how to calculate the probability of a standard normal variable z being between 0 and 1.59 using the cumulative probability from the standard normal table. Step-by-step solution provided with detailed calculations and formula application. Suitable for advanced high school students and anyone studying probability in statistics.

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