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.

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

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!