Find the 10th percentile of x. Round the answer to at least two decimal places.

To find the 10th percentile of \( \bar{x} \), we need to determine the value of \( \bar{x} \) such that the cumulative probability up to that value is 0.10.

### Steps to Solve

1. **Find the z-score corresponding to the 10th percentile:**
   Using the standard normal distribution table, we find that the z-score for the 10th percentile is approximately -1.28.

2. **Convert the z-score to the value of \( \bar{x} \):**
   We use the formula:
   \[
   \bar{x} = \mu + z \times \text{SEM}
   \]
   where \( \mu = 31 \), \( \text{SEM} = 0.7746 \), and \( z = -1.28 \).

3. **Calculate \( \bar{x} \):**
   \[
   \bar{x} = 31 + (-1.28) \times 0.7746
   \]
   \[
   \bar{x} = 31 - 0.9915
   \]
   \[
   \bar{x} \approx 30.01
   \]

### Answer:
The 10th percentile of \( \bar{x} \) is approximately \( 30.01 \).

Do you want details or have any questions? Here are 5 more questions you might be interested in:

1. How to find other percentiles for \( \bar{x} \)?
2. How is the z-score related to percentiles?
3. What is the significance of percentiles in statistics?
4. How to use the normal distribution table for different percentiles?
5. How does the standard error of the mean affect the percentile calculations?

**Tip:** Percentiles are useful for understanding the relative standing of a particular value within a distribution.

Learn how to find the 10th percentile of a dataset using statistical methods. This problem involves calculating the percentile using the z-score formula and understanding its significance in statistics. Suitable for advanced high school students and anyone interested in statistical analysis.

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