Normal Probability Distributions

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 used extensively in statistics to calculate probabilities and critical values for hypothesis testing and confidence intervals.

• Mean (μ): 0

• Standard Deviation (σ): 1

• Notation: Z ~ N(0, 1)

The standard normal distribution is symmetric about the mean, and the total area under the curve is 1.

Z-Scores

A z-score is a standardized value that indicates how many standard deviations a data point is from the mean. Z-scores are used to compare values from different normal distributions and to find probabilities using the standard normal table.

• Formula:

• X: Raw score

• μ: Population mean

• σ: Population standard deviation

Positive z-scores are above the mean, and negative z-scores are below the mean.

Using the Standard Normal Table

The standard normal table (z-table) provides the area (probability) to the left of a given z-score under the standard normal curve. This area represents the cumulative probability up to that z-score.

• To find the probability that Z is less than a value (P(Z < z)), locate the z-score in the table and read the corresponding area.

• To find the probability that Z is greater than a value (P(Z > z)), subtract the table value from 1.

• To find the probability between two z-scores, subtract the smaller area from the larger area.

Example: Finding Probabilities Using the Z-Table

• Example 1: Find P(Z < 1.25).

• Locate 1.2 in the leftmost column and 0.05 in the top row. The intersection gives the area to the left of z = 1.25.

• From the table, P(Z < 1.25) ≈ 0.8944.

• Example 2: Find P(Z > -0.75).

• Find the area to the left of z = -0.75 (from the table, ≈ 0.2266).

• P(Z > -0.75) = 1 - 0.2266 = 0.7734.

Interpreting the Table

The table is typically split into two parts: one for negative z-scores and one for positive z-scores. Each cell gives the cumulative probability from the far left up to the specified z-score.

Additional info: The table values are cumulative probabilities, which are essential for calculating p-values and critical values in hypothesis testing and for constructing confidence intervals.