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The problem involves calculating probabilities using a normal distribution model, \( N(100, 16) \), where the mean (\( \mu \)) is 100 and the variance (\( \sigma^2 \)) is 16. The standard deviation (\( \sigma \)) is therefore \( \sqrt{16} = 4 \).

Let's solve each part step-by-step:

### Part (a): What percent of people's IQs are expected to be over 80?

1. Convert the IQ score of 80 to a z-score:
   \[
   z = \frac{80 - 100}{4} = -5
   \]
   
2. Find \( P(Z > -5) \) from the z-score table. Since this z-score is very low, the probability is almost 1, meaning nearly all people score above 80. 

Thus, about **100%** of people are expected to have an IQ over 80.

---

### Part (b): What percent of people's IQs are expected to be under 90?

1. Convert the IQ score of 90 to a z-score:
   \[
   z = \frac{90 - 100}{4} = -2.5
   \]

2. Find \( P(Z < -2.5) \) from the z-score table. This corresponds to approximately 0.0062.

Thus, about **0.62%** of people are expected to have an IQ under 90.

---

### Part (c): What percent of people's IQs are expected to be between 112 and 124?

1. Convert 112 and 124 to z-scores:
   - For 112:
     \[
     z = \frac{112 - 100}{4} = 3
     \]
   - For 124:
     \[
     z = \frac{124 - 100}{4} = 6
     \]

2. Find \( P(3 < Z < 6) \). Since the upper bound is extremely high, almost 100% of people score below 6. Therefore, we mainly need \( P(Z < 3) \), which is approximately 0.9987.

Thus, about **99.87%** of people have an IQ between 112 and 124.

Would you like further details or have any questions?

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### Related Questions to Explore:
1. How does changing the standard deviation affect the distribution of IQ scores?
2. What percentage of people are expected to have an IQ between 90 and 110?
3. If the mean IQ were 110 instead of 100, how would this affect the results?
4. How would you interpret a negative z-score in this context?
5. How is the probability related to the empirical rule (68-95-99.7 rule) in a normal distribution?

### Tip:
Using a calculator or statistical software can help streamline z-score calculations and improve accuracy for more complex ranges.

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 80?
b) What percent of people's IQs are expected to be under 90?
c) What percent of people's IQs are expected to be between 112 and 124?

Learn how to calculate the percentage of people's IQs over 80, under 90, and between 112 and 124 using the normal distribution model N(100,16). This guide covers key concepts in probability, including Z-scores and the empirical rule, suitable for high school students in Grades 10-12.

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