Use the Normal model ​N(100,16) describing IQ scores to answer the following.
​a) What percent of​ people's IQs are expected to be over 75​?
​b) What percent of​ people's IQs are expected to be under 85?
​c) What percent of​ people's IQs are expected to be between 112 and 132

To solve these questions using the Normal model \( N(100, 16) \), where the mean \( \mu = 100 \) and the standard deviation \( \sigma = 4 \) (since the variance is 16, the standard deviation is the square root of 16).

### a) What percent of people's IQs are expected to be over 75?

1. **Calculate the Z-score for 75**:
   \[
   Z = \frac{X - \mu}{\sigma} = \frac{75 - 100}{4} = -6.25
   \]

2. **Look up the Z-score**:
   A Z-score of -6.25 is extremely low, indicating that virtually all IQ scores are above 75. Thus, the percentage of people with IQs over 75 is approximately 100%.

### b) What percent of people's IQs are expected to be under 85?

1. **Calculate the Z-score for 85**:
   \[
   Z = \frac{85 - 100}{4} = -3.75
   \]

2. **Look up the Z-score**:
   A Z-score of -3.75 is also very low. The percentage of people with IQs under 85 is approximately 0.0001% (essentially 0%).

### c) What percent of people's IQs are expected to be between 112 and 132?

1. **Calculate the Z-scores for 112 and 132**:
   - For 112:
   \[
   Z_{112} = \frac{112 - 100}{4} = 3
   \]

   - For 132:
   \[
   Z_{132} = \frac{132 - 100}{4} = 8
   \]

2. **Look up the Z-scores**:
   - The percentage of people with IQs below 112 corresponds to approximately 99.87% (Z = 3).
   - A Z-score of 8 is extremely high, meaning virtually all scores fall below this, essentially 100%.

3. **Calculate the percentage between 112 and 132**:
   \[
   \text{Percent between 112 and 132} = 100\% - 99.87\% \approx 0.13\%
   \]

### Summary of Results:
- **a)** Approximately 100% of people's IQs are over 75.
- **b)** Approximately 0% of people's IQs are under 85.
- **c)** Approximately 0.13% of people's IQs are between 112 and 132.

If you need further details or clarifications on these calculations, feel free to ask! Here are some related questions to consider:

1. How do you interpret Z-scores in the context of IQ scores?
2. What assumptions does the Normal model make about the distribution of IQ scores?
3. How does the standard deviation affect the spread of IQ scores in a Normal distribution?
4. What are the implications of having a higher mean IQ score in a population?
5. Can you explain how to calculate percentiles using the Normal distribution? 

**Tip:** Always ensure the standard deviation is correctly calculated from the variance before finding Z-scores.

Use the Normal model N(100,16) describing IQ scores to answer the following. a) What percent of people's IQs are expected to be over 75? b) What percent of people's IQs are expected to be under 85? c) What percent of people's IQs are expected to be between 112 and 132?

Explore how to calculate the expected percentages of IQ scores over 75, under 85, and between 112 and 132 using the Normal model N(100,16). Key calculations include Z-scores and their implications. Suitable for advanced statistics students in Grades 11-12.

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