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The problem uses the Normal distribution model \( N(100, 16) \), meaning the mean (average) IQ is 100, and the standard deviation is 16. To answer each question, we will standardize the values using the Z-score formula:

\[
Z = \frac{X - \mu}{\sigma}
\]

where:
- \( X \) is the IQ score,
- \( \mu = 100 \) is the mean,
- \( \sigma = 16 \) is the standard deviation.

### Solution

#### a) What percent of people’s IQs are expected to be over 85?

1. Calculate the Z-score for \( X = 85 \):
   \[
   Z = \frac{85 - 100}{16} = \frac{-15}{16} \approx -0.9375
   \]
2. Using the Z-table, a Z-score of \(-0.94\) corresponds to a cumulative probability of approximately 0.1747. This is the percentage of people with IQs below 85.
3. To find the percentage above 85, subtract from 1:
   \[
   1 - 0.1747 = 0.8253
   \]
   Therefore, approximately **82.5%** of people are expected to have an IQ above 85.

#### b) What percent of people’s IQs are expected to be under 90?

1. Calculate the Z-score for \( X = 90 \):
   \[
   Z = \frac{90 - 100}{16} = \frac{-10}{16} = -0.625
   \]
2. From the Z-table, a Z-score of \(-0.63\) gives a cumulative probability of about 0.2643.
   Thus, approximately **26.4%** of people are expected to have an IQ below 90.

#### c) What percent of people’s IQs are expected to be between 116 and 136?

1. Calculate the Z-score for \( X = 116 \):
   \[
   Z = \frac{116 - 100}{16} = \frac{16}{16} = 1
   \]
   The cumulative probability for \( Z = 1 \) is approximately 0.8413.

2. Calculate the Z-score for \( X = 136 \):
   \[
   Z = \frac{136 - 100}{16} = \frac{36}{16} = 2.25
   \]
   The cumulative probability for \( Z = 2.25 \) is approximately 0.9878.

3. To find the probability between 116 and 136, subtract the probability at \( Z = 1 \) from that at \( Z = 2.25 \):
   \[
   0.9878 - 0.8413 = 0.1465
   \]
   So, approximately **14.7%** of people are expected to have an IQ between 116 and 136.

### Summary of Answers
- a) Approximately **82.5%**
- b) Approximately **26.4%**
- c) Approximately **14.7%**

Would you like further details on any step or have additional questions?

### Related Questions
1. How is the Z-score calculated for values above or below the mean?
2. What does the cumulative probability represent in a normal distribution?
3. Why do we subtract the cumulative probability from 1 to find the "above" probability?
4. What is the empirical rule and how does it apply to IQ scores?
5. How can we use the Z-score to find percentiles in a normal distribution?

### Tip
For quick estimations in a normal distribution, remember that about 68% of data lies within one standard deviation, 95% within two, and 99.7% within three.

Use the Normal model N(100,16) describing IQ scores to answer the following:

1. What percent of people's IQs are expected to be over 85?
2. What percent of people's IQs are expected to be under 90?
3. What percent of people's IQs are expected to be between 116 and 136?

Solve for the percentages of IQ scores above 85, below 90, and between 116 and 136 using the Normal model N(100,16). This problem involves calculating probabilities based on Z-scores and understanding normal distribution properties, suitable for college-level statistics.

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