Finding a z-Score In Exercises 1–16, use the Standard Normal Table or technology to find the z-score that corresponds to the cumulative area or percentile.


P1.5

Verified step by step guidance

1. Identify the cumulative area or percentile given in the problem. For example, if the problem specifies P1.5, this means the cumulative area is 0.015 (1.5%).
2. Understand that the cumulative area represents the area under the standard normal curve to the left of the z-score. This is the probability that a randomly selected value is less than the z-score.
3. Use the Standard Normal Table (Z-table) or technology (such as a statistical calculator or software) to find the z-score corresponding to the cumulative area of 0.015. In the Z-table, locate the value closest to 0.015 in the body of the table and find the corresponding z-score from the row and column headers.
4. If using technology, input the cumulative area (0.015) into the appropriate function or tool (e.g., the inverse cumulative distribution function for the standard normal distribution) to directly obtain the z-score.
5. Verify the z-score obtained by checking that the cumulative area to the left of the z-score matches the given value (0.015). This ensures accuracy in your calculation or lookup.

Key Concepts

Z-Score

A z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It indicates how many standard deviations an element is from the mean. A positive z-score means the value is above the mean, while a negative z-score indicates it is below. Z-scores are essential for standardizing scores on different scales, allowing for comparison across different datasets.

Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution where the mean is 0 and the standard deviation is 1. It is represented by the bell-shaped curve and is used to determine probabilities and percentiles for normally distributed data. The area under the curve corresponds to probabilities, making it a fundamental concept in statistics for understanding how data is distributed.

Cumulative Area

Cumulative area refers to the total area under the curve of a probability distribution up to a certain point. In the context of the standard normal distribution, it represents the probability that a randomly selected score will fall below a specific z-score. This concept is crucial for interpreting z-scores in terms of percentiles, allowing statisticians to understand the relative standing of a score within a distribution.