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This question is about using the Normal distribution to calculate probabilities related to IQ scores, which are assumed to be normally distributed with a mean (\(\mu\)) of 100 and a standard deviation (\(\sigma\)) of 16. The distribution can be represented as \(N(100, 16^2)\).

Let's go through each part of the question:

### Part a) 
We want to find the percentage of people with IQ scores over 85.

1. **Standardize the score**: 
   \[
   Z = \frac{X - \mu}{\sigma} = \frac{85 - 100}{16} = -0.9375
   \]

2. **Find the probability**: Use the standard normal table or a calculator to find \(P(Z > -0.9375)\). This gives us the percentage of people with an IQ score above 85.

### Part b) 
We want to find the percentage of people with IQ scores below 90.

1. **Standardize the score**: 
   \[
   Z = \frac{90 - 100}{16} = -0.625
   \]

2. **Find the probability**: Look up \(P(Z < -0.625)\), which gives the percentage of people with an IQ below 90.

### Part c) 
We want to find the percentage of people with IQ scores between 120 and 136.

1. **Standardize both scores**:
   - For 120: \( Z = \frac{120 - 100}{16} = 1.25 \)
   - For 136: \( Z = \frac{136 - 100}{16} = 2.25 \)

2. **Find the probability**: 
   - Use the standard normal table to find \(P(Z < 2.25)\) and \(P(Z < 1.25)\).
   - The probability for scores between 120 and 136 is \(P(1.25 < Z < 2.25) = P(Z < 2.25) - P(Z < 1.25)\).

Would you like further assistance with calculations or explanations for any part?

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### Related Questions
1. What percentage of people have IQ scores above 115?
2. What is the probability that a randomly selected person has an IQ between 85 and 115?
3. How would the results change if the standard deviation were 20 instead of 16?
4. What IQ score marks the top 5% of the population?
5. How does a normal distribution differ from a binomial distribution?

### Tip:
When working with normal distributions, converting scores to \(Z\)-scores makes it easier to find probabilities using standard normal tables or calculators.

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 85? 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 120 and 136?

Learn how to calculate probabilities for IQ scores using the normal distribution model N(100,16). This tutorial covers calculating the percentage of people with IQs over 85, under 90, and between 120 and 136. Suitable for high school students in Grades 10-12.

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