Assume that adults have IQ scores that are normally distributed with a mean of 100 and a standard deviation of 15, find the IQ score separating the bottom 20% from the top 80%.

The problem asks for the IQ score that separates the bottom 20% from the top 80% in a normally distributed population with a mean of 100 and a standard deviation of 15.

To solve this, we need to find the z-score corresponding to the cumulative probability of 20% in the standard normal distribution, then convert that z-score to the IQ scale.

### Steps:
1. **Identify the cumulative probability**: The cumulative probability is 0.20 (since 20% is below the cutoff).
   
2. **Find the corresponding z-score**: We need to find the z-score associated with the cumulative probability of 0.20. Using z-tables or a statistical tool, we find that the z-score for 0.20 is approximately **-0.8416**.

3. **Convert z-score to IQ score**: Use the formula that relates the z-score to the IQ score:
   \[
   x = \mu + z \cdot \sigma
   \]
   where:
   - \( x \) is the IQ score we're trying to find,
   - \( \mu = 100 \) is the mean,
   - \( \sigma = 15 \) is the standard deviation,
   - \( z = -0.8416 \).

   Substituting the values:
   \[
   x = 100 + (-0.8416) \cdot 15
   \]
   \[
   x = 100 - 12.624 = 87.38
   \]

### Final answer:
The IQ score that separates the bottom 20% from the top 80% is approximately **87.38**.

Would you like further details on this solution or any clarifications?

### Related questions:
1. What is the IQ score separating the bottom 10% from the top 90%?
2. How can you calculate the z-score for other cumulative probabilities?
3. What IQ score corresponds to the top 1% in this distribution?
4. How does changing the mean or standard deviation affect the distribution?
5. How do you interpret a z-score in a normal distribution?

**Tip**: To quickly find z-scores for various cumulative probabilities, consider using a standard normal distribution table or statistical calculator.

Learn how to find the IQ score that separates the bottom 20% from the top 80% in a normally distributed population. This problem uses key statistical concepts, such as the standard normal distribution and z-scores, with a step-by-step solution. Ideal for students in Grades 10-12.

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