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To answer the questions, we will use the normal distribution \( N(100, 16) \), where the mean is 100 and the standard deviation is 16. Here's how we'll approach each part:

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

First, we calculate the z-score for an IQ of 80:
\[
z = \frac{80 - 100}{16} = \frac{-20}{16} = -1.25
\]
We then look up the cumulative probability for \( z = -1.25 \) from standard normal distribution tables or use a calculator. The cumulative probability corresponding to \( z = -1.25 \) is approximately 0.1056. This means about 10.56% of people have IQs below 80. So the percent of people with IQs above 80 is:
\[
100\% - 10.56\% = 89.44\%
\]

Thus, approximately 89.4% of people’s IQs are expected to be above 80.

---

### b) What percent of people's IQs are expected to be under 95?

Now, we calculate the z-score for 95:
\[
z = \frac{95 - 100}{16} = \frac{-5}{16} = -0.3125
\]
The cumulative probability for \( z = -0.3125 \) is approximately 0.3770, meaning about 37.70% of people have IQs below 95.

Thus, approximately **37.7%** of people’s IQs are expected to be below 95.

---

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

1. For an IQ of 116, the z-score is:
\[
z = \frac{116 - 100}{16} = \frac{16}{16} = 1
\]
The cumulative probability for \( z = 1 \) is approximately 0.8413.

2. For an IQ of 136, the z-score is:
\[
z = \frac{136 - 100}{16} = \frac{36}{16} = 2.25
\]
The cumulative probability for \( z = 2.25 \) is approximately 0.9878.

Now, subtract the cumulative probabilities:
\[
0.9878 - 0.8413 = 0.1465
\]

Thus, approximately **14.7%** of people’s IQs are expected to be between 116 and 136.

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Would you like more details on any of these steps, or clarification?

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Here are five related questions:
1. How is the z-score formula derived for normal distributions?
2. What does it mean for a distribution to be "normal"?
3. How do you interpret cumulative probability values?
4. How do percentiles relate to the z-score in normal distributions?
5. What are the applications of normal distribution in real-world statistics?

**Tip:** Always double-check cumulative probabilities with a standard normal distribution table or calculator!

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 95?
c) What percent of people's IQs are expected to be between 116 and 136?

Learn how to calculate the probability of different IQ score ranges using a normal distribution model with a mean of 100 and a standard deviation of 16. Understand how to apply the Z-score formula to find probabilities for scores above 80, below 95, and between 116 and 136.

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