The question asks for calculations using the normal distribution, specifically $N(100, 16)$, where the mean IQ is 100, and the standard deviation is 16.

We need to calculate the following probabilities:

1. Percentage of people with IQs above 85.
2. Percentage of people with IQs below 95.
3. Percentage of people with IQs between 112 and 136.

Step 1: Standardize the IQ scores 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.

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

1. Find the Z-score for 85: $Z = \frac{85 - 100}{16} = \frac{-15}{16} = -0.9375$

2. Use a Z-table or calculator to find the probability for $Z = -0.94$, which gives approximately 0.174.

3. To find the percentage above 85: $1 - 0.174 = 0.826$ So, about 82.6% of people have IQs above 85.

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

1. Find the Z-score for 95: $Z = \frac{95 - 100}{16} = \frac{-5}{16} = -0.3125$
2. Using a Z-table or calculator, the probability for $Z = -0.31$ is approximately 0.378.

So, about 37.8% of people have IQs below 95.

c) What percent of people's IQs are expected to be between 112 and 136?

1. Find the Z-score for 112: $Z = \frac{112 - 100}{16} = \frac{12}{16} = 0.75$ Using a Z-table, the probability for $Z = 0.75$ is approximately 0.773.

2. Find the Z-score for 136: $Z = \frac{136 - 100}{16} = \frac{36}{16} = 2.25$ Using a Z-table, the probability for $Z = 2.25$ is approximately 0.9878.

3. To find the probability between 112 and 136: $0.9878 - 0.773 = 0.2148$ So, about 21.5% of people have IQs between 112 and 136.

Let me know if you'd like more details on any part!

Here are 5 related questions to expand on this:

1. How do you calculate Z-scores for any given distribution?
2. What does a negative Z-score represent?
3. How does changing the standard deviation affect the shape of the normal curve?
4. What is the total area under the normal distribution curve?
5. How can you use technology (like a calculator or software) to find these probabilities?

Tip: Always round Z-scores and probabilities to a reasonable number of decimal places for precision.