you found out that the requirement is too stringent and too many people are getting excluded.  You need to change the requirement so only the bottom 8% and the top 12% are excluded.  What should the new BMIs be?

To find the new BMI cutoffs, we need to exclude the bottom 8% and the top 12% of the population. Assuming that BMI follows a normal distribution, we can use the corresponding z-scores for these percentiles.

### Step-by-Step Explanation:
1. **Percentiles for BMI:**
   - We want to exclude the bottom 8% and the top 12%.
   - The bottom 8% corresponds to the **8th percentile**.
   - The top 12% corresponds to the **88th percentile** (100% - 12%).

2. **Find the z-scores for these percentiles:**
   - For the **8th percentile**, the z-score is approximately -1.41.
   - For the **88th percentile**, the z-score is approximately 1.18.

3. **Convert z-scores to BMI values:**
   To calculate the actual BMI values, we need the mean (\( \mu \)) and standard deviation (\( \sigma \)) of the BMI distribution. These are usually estimated from population data. Assuming we know these values, we can apply the formula for converting z-scores to BMI:

   \[
   \text{BMI} = \mu + (z \cdot \sigma)
   \]

   - For the **8th percentile**, the BMI value is:
     \[
     \text{BMI}_{8\%} = \mu + (-1.41 \cdot \sigma)
     \]

   - For the **88th percentile**, the BMI value is:
     \[
     \text{BMI}_{88\%} = \mu + (1.18 \cdot \sigma)
     \]

Without the exact values of \( \mu \) (mean BMI) and \( \sigma \) (standard deviation of BMI), we can't calculate the exact BMI cutoffs, but the above z-scores tell us how far below and above the mean these cutoffs will be.

If you provide the mean and standard deviation of the BMI distribution, I can give you the precise BMI cutoffs.

Would you like more details on z-scores, or do you have any other questions?

Here are 5 related questions you might find helpful:
1. How do z-scores relate to normal distributions?
2. What is the mean and standard deviation of BMI in a typical population?
3. How does changing the exclusion criteria affect the cutoff percentages?
4. How can I use z-scores to interpret other health metrics?
5. What is the relationship between percentiles and probability?

**Tip:** The z-score is a standardized value that helps compare data points from different distributions by expressing them in terms of standard deviations from the mean.

You found out that the requirement is too stringent and too many people are getting excluded. You need to change the requirement so only the bottom 8% and the top 12% are excluded. What should the new BMIs be?

Learn how to calculate new BMI cutoffs that exclude the bottom 8% and top 12% using z-scores in a normal distribution. This step-by-step guide explains how to convert z-scores to BMI using the formula BMI = μ + (z × σ) and the properties of percentiles. Suitable for high school or college-level students.

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