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.


P91

Verified step by step guidance

1. Understand the problem: The goal is to find the z-score that corresponds to the 91st percentile (P91). This means we are looking for the z-score where 91% of the data lies below it in a standard normal distribution.
2. Recall that the standard normal distribution has a mean (μ) of 0 and a standard deviation (σ) of 1. The cumulative area under the curve to the left of the z-score represents the percentile.
3. Use the Standard Normal Table (Z-table) or technology to find the z-score. Locate the cumulative area closest to 0.91 in the Z-table. The corresponding z-score is the value you are looking for.
4. If using technology (e.g., a calculator or statistical software), use the inverse cumulative distribution function (often denoted as invNorm or similar). Input the cumulative area (0.91), the mean (0), and the standard deviation (1) to calculate the z-score.
5. Interpret the result: The z-score you find represents the number of standard deviations above the mean where the 91st percentile lies in the standard normal distribution.

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 indicates 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.