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.