Lecture 22: The Normal Distribution – Areas, Probabilities, and Percentiles

Normal Probability Distributions

Introduction to the Standard Normal Distribution

The normal distribution is a fundamental continuous probability distribution in statistics, characterized by its bell-shaped curve. The standard normal distribution is a special case with mean 0 and standard deviation 1. Probabilities and areas under the curve are often computed using z-scores and standard normal tables (Z-tables).

• Z-score: The number of standard deviations a value is from the mean. Calculated as $z = \frac{x - \mu}{\sigma}$, where $x$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation.

• Standard Normal Table (Z-table): Provides the area (probability) to the left of a given z-score under the standard normal curve.

Computing Areas Under the Normal Curve

Area to the Left of a Z-score

To find the area under the standard normal curve to the left of a specific z-score:

• Use the Z-table to find the area corresponding to the z-score.

• Example: Area to the left of $z = -0.38$ is 0.3520.

Area to the Right of a Z-score

To find the area to the right of a specific z-score:

• Subtract the area to the left from 1: $P(Z > z) = 1 - P(Z < z)$.

• Example: Area to the right of $z = 1.25$ is $1 - 0.8944 = 0.1056$.

Area Between Two Z-scores

To find the area between two z-scores $z_1$ and $z_2$:

• Subtract the area to the left of $z_1$ from the area to the left of $z_2$:

$P(z_1 < Z < z_2) = P(Z < z_2) - P(Z < z_1)$

• Example: Area between $z = -1.02$ and $z = 2.94$ is $0.9984 - 0.1539 = 0.8445$.

Generalization to Any Normal Random Variable

• For any normal random variable $X \sim N(\mu, \sigma^2)$, convert $x$ to a z-score using $z = \frac{x - \mu}{\sigma}$.

• Use the Z-table to find the area to the left, right, or between values as above.

Finding Cutoff Points (Percentiles and Probabilities)

Procedure for Finding a Value Corresponding to a Given Probability

1. Draw a normal curve and shade the area corresponding to the given probability, proportion, or percentile.

2. Use the Z-table to find the z-score that corresponds to the shaded area.

3. Convert the z-score back to the original value using $x = \mu + z\sigma$.

Example: For body temperatures with $\mu = 98.2^\circ F$ and $\sigma = 0.73^\circ F$, the cutoff for the lowest 3% is found by solving $P(X < x) = 0.03$, which corresponds to $z = -1.88$. Thus, $x = 98.2 + (-1.88)(0.73) = 96.8^\circ F$.

Properties of Continuous Random Variables

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

• Thus, $P(a < X < b) = P(a \leq X < b) = P(a < X \leq b) = P(a \leq X \leq b)$.

Computing Normal Probabilities: Step-by-Step

Step 1: Convert to Z-scores

• For $X \sim N(\mu, \sigma^2)$, convert $a$ and $b$ to z-scores: $z_a = \frac{a - \mu}{\sigma}$, $z_b = \frac{b - \mu}{\sigma}$.

Step 2: Use the Z-table

• Find the probabilities for the corresponding z-scores using the Z-table.

Remarks on Using the Z-table

• The Z-table typically covers $-3.40 \leq z \leq 3.40$.

• Table entries give $P(Z < z)$, the area to the left of $z$.

• Z-scores are given to two decimal places: the first column gives the ones and tenths, the first row gives the hundredths.

• Example: For $z = 1.68$, find 1.6 in the first column and 0.08 in the first row.

Types of Normal Probabilities

• Left-tail: $P(Z < z)$ – direct from the table.

• Right-tail: $P(Z > z) = 1 - P(Z < z)$ or $P(Z > z) = P(Z < -z)$ (by symmetry).

• Between two values: $P(z_1 < Z < z_2) = P(Z < z_2) - P(Z < z_1)$.

• Absolute value: $P(|Z| > z) = 2P(Z < -z)$.

Example: Computing Normal Probabilities

Let $X$ be a normal random variable with $\mu = 10$, $\sigma = 5$.

• (a) $P(X \leq 12)$: $z = \frac{12-10}{5} = 0.40$, $P(Z \leq 0.40) = 0.6554$

• (b) $P(X > 12)$: $P(Z > 0.40) = 1 - 0.6554 = 0.3446$

• (c) $P(12 \leq X \leq 15)$: $z_1 = 0.40$, $z_2 = 1.00$, $P(0.40 \leq Z \leq 1.00) = 0.8413 - 0.6554 = 0.1859$

• (d) $P(|X| \leq 15)$: $z = 1.00$, $P(-1.00 \leq Z \leq 1.00) = 0.8413 - 0.1587 = 0.6826$

Percentiles of the Normal Distribution

Definition and Calculation

The p-th percentile $x_p$ of a continuous random variable $X$ is the value such that $P(X \leq x_p) = p$.

• For the standard normal, find $z_p$ such that $P(Z \leq z_p) = p$.

• If $p$ is not in the table, use the closest value or average the two closest z-scores.

Examples

• 95th percentile: $P(Z \leq z_0) = 0.95$; $z_0 = 1.645$ (average of 1.64 and 1.65 if needed).

• Application: For $X \sim N(40, 36)$, $\sigma = 6$:

  • 95th percentile: $x_0 = 40 + 6(1.645) = 49.87$

  • 5th percentile: $x_0 = 40 + 6(-1.645) = 30.13$

  • Find $x_0$ such that $P(X \geq x_0) = 0.7995$: $z_0 = -0.84$, $x_0 = 40 + 6(-0.84) = 34.96$

Summary Table: Types of Normal Probabilities

Probability Type	Formula	How to Find
Left-tail	$P(Z < z)$	Direct from Z-table
Right-tail	$P(Z > z) = 1 - P(Z < z)$	1 minus Z-table value
Between two values	$P(z_1 < Z < z_2) = P(Z < z_2) - P(Z < z_1)$	Subtract Z-table values
Absolute value	$P(|Z| > z) = 2P(Z < -z)$	Double left-tail for negative z

Key Takeaways

• Always convert to z-scores before using the Z-table.

• For percentiles, use the Z-table in reverse: find the z-score for the given area, then convert back to the original scale.

• For continuous random variables, the probability of a single value is zero.