Finding z-Scores from Areas Under the Normal Curve

Reverse Lookup: Area to z-Score

In normal distribution problems, we often need to find the area under the curve for a given z-score. However, sometimes the reverse is required: given an area, determine the corresponding z-score. This process is essential for understanding percentiles, critical values, and hypothesis testing.

• Area to the Left: The standard normal table (z-table) provides the area to the left of a given z-score. To find the z-score for a specified area, scan the table for the closest area value and read the corresponding z-score.

• Example: To find the z-score with an area of 0.93 to its left, look for 0.9300 in the table. If the exact value is not present, choose the closest (e.g., 0.9306 corresponds to z = 1.48).

• Percentiles: The z-score for the 93rd percentile is the value with 93% of the area to its left.

• Area to the Right: If the area is given to the right, subtract from 1 to convert to the left. For example, an area of 0.07 to the right corresponds to 1 - 0.07 = 0.93 to the left.

• Ties: If two table values are equally close, average their z-scores.

Formula:

Additional info: The process is used for finding percentiles, critical values, and in hypothesis testing.

Using Technology: Calculator Functions

Modern calculators and statistical software provide functions to find z-scores from areas directly. The TI calculator's invNorm function is commonly used.

• invNorm(p): Returns the z-score with area p to its left.

• Rounding: Round z-scores to two decimal places for consistency with tables.

• Accuracy: Calculators are more precise than tables, but either method is acceptable for homework and exams.

Example: Using invNorm(0.93) yields z ≈ 1.48.

Critical Values and zα Notation

Definition and Application

Critical values are z-scores that correspond to specific areas in the tail(s) of the normal distribution. The notation zα is used to denote the z-score with area α to its right.

• zα: The z-score with area α to its right.

• Finding zα: Use invNorm(1 - α) to find the value.

• Symmetry: The normal distribution is symmetric, so invNorm(α) yields the negative of the desired z-score.

• Small α: α is typically a small area, representing the probability of observing an extreme value.

Examples:

• For α = 0.10: invNorm(0.90) gives z = 1.28.

• For α = 0.04: invNorm(0.96) gives z = 1.75.

Definition:

Percentiles and Population Separation

Percentiles and population separation questions can be translated into area-to-z-score problems.

• Percentile: The z-score for the nth percentile is the value with n% area to its left.

• Population Separation: The z-score separating the bottom x% from the rest is the value with x% area to its left.

Example: The z-score separating the top 7% from the rest is the value with 93% area to its left.

Summary Table: Area and z-Score Relationships

Additional info: This process is fundamental for constructing confidence intervals, performing hypothesis tests, and interpreting statistical results in the context of the normal distribution.