Standard Normal Distribution and Using the z-Table

Standard Normal Distribution

Introduction to the Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution with a mean of 0 and a standard deviation of 1. It is widely used in statistics to calculate probabilities and to standardize values using z-scores. The area under the curve represents probabilities associated with different z-scores.

• Mean (μ): 0

• Standard deviation (σ): 1

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

Definition of z-score

A z-score is calculated as:

$z = \frac{X - \mu}{\sigma}$

• X: Observed value

• μ: Mean of the distribution

• σ: Standard deviation of the distribution

Using the z-Table to Find Probabilities

Finding Standard Normal Probabilities Using the z-Table

The z-table provides cumulative probabilities (areas under the curve) to the left of a given z-score. To find probabilities:

1. Sketch the normal curve and shade the area of interest.

2. Look up the z-score in the z-table to find the corresponding area.

3. If the area to the right is needed, subtract the table value from 1.

Key Points

• Area to the Left: $P(Z < z)$ is found directly from the z-table.

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

• Area Between Two z-values: $P(z_1 < Z < z_2) = P(Z < z_2) - P(Z < z_1)$

Example

• To find $P(Z < -0.64)$, look up -0.64 in the z-table: area = 0.2611.

• To find $P(Z > 2.77)$, look up 2.77 in the z-table: area = 0.9972. Then, $1 - 0.9972 = 0.0028$.

• To find $P(-0.64 < Z < 2.77)$, subtract: $0.9972 - 0.2611 = 0.7361$.

Finding Probabilities Between Two z-Scores

Procedure

To find the probability between two z-scores:

1. Find the area to the left of the larger z-score.

2. Find the area to the left of the smaller z-score.

3. Subtract the smaller area from the larger area.

Example

• Find $P(-0.64 < Z < 2.77)$:

• Area to left of 2.77: 0.9972

• Area to left of -0.64: 0.2611

• Probability: $0.9972 - 0.2611 = 0.7361$

Finding z-Scores from Probabilities

Procedure

To find the z-score corresponding to a given probability:

1. Identify the area (probability) under the curve.

2. Look up the area in the z-table to find the corresponding z-score.

3. If the area is to the right, subtract the probability from 1 before using the z-table.

Example

• Given area to the left is 0.8631, find the z-score:

• Look up 0.8631 in the z-table: $z = 1.10$

• If area to the right is 0.0670, first calculate $1 - 0.0670 = 0.9330$, then look up 0.9330 in the z-table: $z = 1.50$

Practice Problems

• Find the area under the standard normal distribution to the left of a z-score of 1.21: $P(Z < 1.21) = 0.8869$

• Find the area under the standard normal distribution to the right of a z-score of -0.64: $P(Z > -0.64) = 1 - 0.2611 = 0.7389$

• Find the z-score such that $P(Z < z) = 0.8631$: $z = 1.10$

• Find the z-score corresponding to an area of 0.91557: $z = 1.37$ (approximate)

z-Table Reference

The z-table lists cumulative probabilities for z-scores, typically from -3.4 to +3.4. Values represent the area under the standard normal curve to the left of the given z-score.

z-score	Area to Left
-0.64	0.2611
1.21	0.8869
2.77	0.9972
1.10	0.8631
1.37	0.9157

Additional info: The z-table is essential for finding probabilities and critical values in hypothesis testing and confidence interval estimation.