Section 7.2 Applications of the Normal Distribution: Area and Value Calculations

Section 7.2: Applications of the Normal Distribution

Objective 1: Find and Interpret the Area under a Normal Curve

The normal distribution is a fundamental probability distribution in statistics, often used to model real-world phenomena. Understanding how to find and interpret areas under the normal curve is essential for calculating probabilities and percentiles.

• Standard Normal Distribution: The standard normal distribution is a normal distribution with mean $\mu = 0$ and standard deviation $\sigma = 1$. Any normal random variable $X$ can be transformed to the standard normal variable $Z$ using the formula: $Z = \frac{X - \mu}{\sigma}$

• Area under the Curve: The area under the normal curve between two values represents the probability that the random variable falls within that interval.

• Using Z-Tables: Z-tables (Table V) are used to find the area (probability) to the left of a given Z-score.

Example: Finding and Interpreting Area Under a Normal Curve

A pediatrician studies the heights of 200 three-year-old female patients. The heights are approximately normally distributed, with mean 34.5 inches and standard deviation 3.17 inches. To find the proportion of 3-year-old females taller than 31 inches:

• Calculate the Z-score for 31 inches: $Z = \frac{31 - 34.5}{3.17}$

• Use the Z-table to find the area to the right of this Z-score.

• This area represents the proportion of girls taller than 31 inches.

Additional info: The area to the left of a Z-score gives the cumulative probability up to that value.

Summary of Methods for Finding Areas

• Area to the Left of z: Use the Z-table directly to find $P(Z < z)$.

• Area to the Right of z: $P(Z > z) = 1 - P(Z < z)$

• Area Between $z_1$ and $z_2$: $P(z_1 < Z < z_2) = P(Z < z_2) - P(Z < z_1)$

Objective 2: Find the Value of a Normal Random Variable

Sometimes, we need to find the value of a normal random variable that corresponds to a specific percentile or probability. This involves working backwards from the area under the curve to the value of $X$.

• Percentiles: The $k$th percentile is the value below which $k$ percent of the data fall.

• Finding $X$ from a Percentile: Use the Z-table to find the Z-score corresponding to the desired percentile, then solve for $X$ using: $X = \mu + Z\sigma$

Example: Finding the Value of a Normal Random Variable

Suppose the heights of 3-year-old females are normally distributed with mean 34.5 inches and standard deviation 3.17 inches. To find the height at the 20th percentile:

• Find the Z-score for the 20th percentile using the Z-table (look for area closest to 0.20).

• Calculate $X = 34.5 + Z \times 3.17$

Notation $x_k$

• $x_k$: Denotes the value of the random variable at the $k$th percentile.

Probability of a Specific Value

• For any continuous random variable, the probability of observing any specific value is zero: $P(X = x) = 0$

• This is because the area under a single point on a continuous curve is zero.

Summary Table: Methods for Finding Areas under the Normal Curve

Type of Area	Method	Formula
Left of $z$	Use Z-table	$P(Z < z)$
Right of $z$	Subtract from 1	$P(Z > z) = 1 - P(Z < z)$
Between $z_1$ and $z_2$	Subtract left areas	$P(z_1 < Z < z_2) = P(Z < z_2) - P(Z < z_1)$

Key Terms

• Normal Distribution: A symmetric, bell-shaped distribution defined by mean $\mu$ and standard deviation $\sigma$.

• Z-score: The number of standard deviations a value is from the mean.

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

• Continuous Random Variable: A variable that can take any value within a range.