 What percent of​ people's IQs are expected to be between 116 and 128​?The normal model is 100,16

Let's solve the problem where we need to find the percentage of people's IQs expected to be between 116 and 128, assuming a normal distribution with a mean \( \mu = 100 \) and a standard deviation \( \sigma = 16 \).

### Step-by-step Solution:

1. **Convert the raw scores to z-scores:**
   The z-score formula is:

   \[
   z = \frac{x - \mu}{\sigma}
   \]

   Where:
   - \( x \) is the raw score (either 116 or 128),
   - \( \mu = 100 \) (mean of IQ distribution),
   - \( \sigma = 16 \) (standard deviation of IQ distribution).

   - For \( x = 116 \):
     \[
     z_1 = \frac{116 - 100}{16} = \frac{16}{16} = 1.0
     \]

   - For \( x = 128 \):
     \[
     z_2 = \frac{128 - 100}{16} = \frac{28}{16} = 1.75
     \]

2. **Find the cumulative probabilities for these z-scores:**
   Using a standard normal distribution table (or a calculator):
   
   - For \( z_1 = 1.0 \), the cumulative probability is approximately 0.8413 (or 84.13%).
   - For \( z_2 = 1.75 \), the cumulative probability is approximately 0.9599 (or 95.99%).

3. **Calculate the percentage between these two z-scores:**

   \[
   \text{Percentage} = (0.9599 - 0.8413) = 0.1186 = 11.86\%
   \]

Thus, **11.86%** of people's IQs are expected to be between 116 and 128.

Would you like additional clarification on any step?

### 5 Related Questions:
1. How can we calculate the probability of a person's IQ being greater than 130 in this model?
2. How does the area under the normal curve relate to probabilities?
3. Can we apply the same method for non-normal distributions?
4. How would changing the standard deviation affect the results?
5. What percentage of people's IQs fall between 85 and 115?

### Tip:
When using a normal distribution, about 68% of data falls within one standard deviation of the mean, and about 95% falls within two standard deviations.

What percent of​ people's IQs are expected to be between 116 and 128​? The normal model is 100,16

Learn how to calculate the percentage of people's IQs that fall between 116 and 128 in a normal distribution with a mean of 100 and standard deviation of 16. This problem covers key statistical concepts, including Z-scores and cumulative probabilities. Suitable for students in Grades 9-12.

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