Normal Distribution and Probability Guidance for Chapter 6

Q8. Find the area of the shaded region. The graph depicts IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler test).

Background

Topic: Normal Distribution, Probability as Area Under the Curve

This question tests your ability to interpret a normal distribution graph and calculate the probability (area) for a specified region, given the mean and standard deviation.

Key Terms and Formulas:

• Normal Distribution: A symmetric, bell-shaped curve describing the spread of a variable.

• Mean ($\mu$): The center of the distribution (here, $\mu = 100$).

• Standard Deviation ($\sigma$): Measures spread (here, $\sigma = 15$).

• Z-score: $z = \frac{X - \mu}{\sigma}$

• Area under the curve: Represents probability for a range of values.

Step-by-Step Guidance

1. Identify the mean ($\mu = 100$) and standard deviation ($\sigma = 15$) from the problem statement.

2. Examine the graph to determine the range of IQ scores corresponding to the shaded region. (Is it above or below a certain value?)

3. Convert the relevant IQ score(s) to z-scores using $z = \frac{X - \mu}{\sigma}$.

4. Use the z-score to find the area under the normal curve (probability) for the shaded region, either from a z-table or using technology.

Try solving on your own before revealing the answer!

Final Answer: 0.7486

The area of the shaded region is approximately 0.7486, meaning about 74.86% of adults have IQ scores in the shaded range.

Q9. Find the indicated IQ score. The graph depicts IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler test). The shaded area under the curve is 0.5675.

Background

Topic: Normal Distribution, Finding a Value Given a Probability

This question tests your ability to use the normal distribution to find the IQ score corresponding to a given cumulative probability (area under the curve).

Key Terms and Formulas:

• Percentile: The value below which a given percentage of observations fall.

• Z-score: $z = \frac{X - \mu}{\sigma}$

• Inverse Normal Calculation: Find $z$ for a given area, then solve for $X$.

Step-by-Step Guidance

1. Identify the mean ($\mu = 100$) and standard deviation ($\sigma = 15$).

2. Recognize that the shaded area (0.5675) represents the cumulative probability to the left of the unknown IQ score.

3. Use a z-table or technology to find the z-score corresponding to a cumulative probability of 0.5675.

4. Once you have the z-score, use the formula $X = \mu + z\sigma$ to find the corresponding IQ score.

Try solving on your own before revealing the answer!

Final Answer: 102.6

The IQ score corresponding to the cumulative area of 0.5675 is approximately 102.6.