Use the standard normal table to find the​ z-score that corresponds to the given percentile. If the area is not in the​ table, use the entry closest to the area. If the area is halfway between two​ entries, use the​ z-score halfway between the corresponding​ z-scores. If​ convenient, use technology to find the​ z-score.
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To find the \( z \)-score corresponding to a given percentile \( p \) using the standard normal table (z-table), follow these steps:

1. **Locate the closest area**: Find the area closest to \( p \) in the z-table. This area represents the cumulative probability up to that \( z \)-score.
2. **Identify the corresponding \( z \)-score**: The z-table will give you the \( z \)-score corresponding to that cumulative probability.
3. **Handle midway areas**: If the area \( p \) is halfway between two entries in the table, take the average of the two corresponding \( z \)-scores.

Let's find the \( z \)-score for \( p1 \).

Assuming \( p1 \) is a specific percentile given by the user, I'll demonstrate with an example of finding the \( z \)-score for the 90th percentile (p = 0.9000):

1. **Locate the closest area**: In the z-table, look for the area closest to 0.9000.
   - The area 0.8997 corresponds to a \( z \)-score of 1.28.
   - The area 0.9015 corresponds to a \( z \)-score of 1.29.
2. **Identify the \( z \)-score**: Since 0.9000 is between 0.8997 and 0.9015, we take the average of 1.28 and 1.29:
   \[
   z = \frac{1.28 + 1.29}{2} = 1.285
   \]

So, the \( z \)-score corresponding to the 90th percentile is approximately 1.285.

If you have a specific percentile \( p1 \) in mind, please provide that value, and I can find the corresponding \( z \)-score for you.

Do you want more details or have any questions? Here are some related questions for further exploration:

1. What is the \( z \)-score for the 95th percentile?
2. How do you find the \( z \)-score for the 75th percentile?
3. Can you explain how to use the z-table to find the area to the left of a given \( z \)-score?
4. How do you find the percentile corresponding to a \( z \)-score of -1.5?
5. What is the area under the standard normal curve between \( z = 0 \) and \( z = 1 \)?
6. How does the symmetry of the normal distribution affect the calculation of percentiles?
7. Can technology tools like graphing calculators or statistical software help in finding \( z \)-scores more efficiently?
8. What is the \( z \)-score for the 99th percentile?

**Tip**: When working with the z-table, always ensure that the table you're using is standardized for the cumulative distribution function of the normal distribution.

Learn how to find the z-score corresponding to a given percentile using the standard normal table (z-table). This detailed explanation covers locating the closest area in the z-table, handling midway areas, and using technology for accurate results. Suitable for advanced high school students and college-level learners.

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