Assume that adults have IQ scores that are normally distributed with a mean of 20 and a standard deviation of 12. Find the IQ score separating the top 16% from the others.

To find the IQ score that separates the top 16% from the others, we are looking for a cutoff value in a normally distributed population. The steps are as follows:

1. **Given Information:**
   - Mean (μ) = 100
   - Standard deviation (σ) = 12
   - We want the IQ score that separates the top 16% from the others, meaning we're looking for the value corresponding to the 84th percentile (since 100% - 16% = 84%).

2. **Find the z-score for the 84th percentile:**
   We can use standard normal distribution tables or a calculator to find the z-score corresponding to a cumulative probability of 0.84.

   The z-score for 0.84 is approximately **0.995**.

3. **Convert the z-score to an actual IQ score:**
   We can use the formula for a normal distribution:

   \[
   X = \mu + z \cdot \sigma
   \]

   Where:
   - \( \mu = 100 \) (mean)
   - \( z = 0.995 \) (z-score for 84th percentile)
   - \( \sigma = 12 \) (standard deviation)

   Substituting in the values:

   \[
   X = 100 + (0.995 \times 12)
   \]

   \[
   X = 100 + 11.94 = 111.94
   \]

Thus, the IQ score that separates the top 16% from the others is approximately **111.94**.

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Would you like more details on this, or have any questions?

Here are some related questions to expand on this topic:
1. How do you calculate z-scores for different percentiles?
2. What is the probability of having an IQ score above 130 in this distribution?
3. How would the separation point change if the standard deviation were smaller?
4. What is the IQ score that separates the top 10% from the rest?
5. How do you find the area under a normal curve using a z-table?

**Tip:** Always remember to subtract from 100% when you're dealing with the top percentile, as you're looking at the upper tail of the distribution!

Assume that adults have IQ scores that are normally distributed with a mean of 100 and a standard deviation of 12. Find the IQ score separating the top 16% from the others.

Learn how to calculate the IQ score that separates the top 16% from the others in a normally distributed population. This problem covers key concepts such as normal distribution, percentiles, and z-scores, with a mean IQ of 100 and a standard deviation of 12. Suitable for students in Grades 10-12.

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