Complete the statement. Round your answer to the nearest thousandth.
In a population that is normally distributed with mean 92 and standard deviation 12, the bottom 80% of the values are those less than .

To find the value below which the bottom 80% of the values fall in a normally distributed population with a mean of 92 and a standard deviation of 12, you need to find the 80th percentile of this distribution.

1. **Find the z-score corresponding to the 80th percentile:** The z-score for the 80th percentile is approximately 0.841. This z-score represents the number of standard deviations away from the mean.

2. **Use the z-score formula to find the corresponding value:**
   
   \[
   X = \mu + z \cdot \sigma
   \]

   where:
   - \( \mu = 92 \) (mean)
   - \( \sigma = 12 \) (standard deviation)
   - \( z = 0.841 \) (z-score for the 80th percentile)

3. **Calculate the value:**

   \[
   X = 92 + (0.841 \times 12) = 92 + 10.092 = 102.092
   \]

Rounded to the nearest thousandth, the bottom 80% of the values are those less than **102.092**.

Would you like details on the steps? Here are some related questions for further practice:

1. How would you find the value below which the bottom 90% of the values fall?
2. What is the z-score for the 50th percentile in a normal distribution?
3. How does the standard deviation affect the position of the 80th percentile?
4. What would be the 20th percentile for the same population?
5. How do you interpret the 80th percentile in the context of this distribution?

**Tip:** Understanding how to use the z-score table can greatly simplify finding percentiles in a normal distribution.

Learn how to find the value below which the bottom 80% of a normally distributed population with mean 92 and standard deviation 12 fall. Step-by-step solution with calculations using the z-score formula. Understand the concept of percentiles in statistics.

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