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Ch. 5 - Normal Probability Distributions
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLarson 8th EditionCh. 5 - Normal Probability DistributionsProblema 5.3.6
Capitolo 5, Problema 5.3.6

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.


0.94

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Step 1: Understand the problem. You are tasked with finding the z-score that corresponds to a cumulative area (or percentile) of 0.94 under the standard normal distribution. The cumulative area represents the probability that a value is less than or equal to the z-score.
Step 2: Recall that the standard normal distribution is symmetric with a mean of 0 and a standard deviation of 1. The cumulative area of 0.94 means that 94% of the data lies to the left of the z-score.
Step 3: Use the Standard Normal Table (also called the z-table) or technology (such as a statistical calculator or software) to find the z-score. Locate the cumulative area of 0.94 in the table, or use the inverse cumulative distribution function (often denoted as invNorm or similar) in technology.
Step 4: If using the z-table, find the closest value to 0.94 in the body of the table. Then, identify the corresponding z-score by combining the row and column headers. If using technology, input the cumulative area (0.94) into the inverse cumulative function to directly obtain the z-score.
Step 5: Interpret the result. The z-score you find represents the number of standard deviations above the mean where 94% of the data lies to the left. This z-score is positive because 0.94 is greater than 0.5, indicating it is on the right side of the mean.

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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.
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Z-Scores From Given Probability - TI-84 (CE) Calculator

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 symmetric and bell-shaped, representing the distribution of many natural phenomena. The area under the curve corresponds to probabilities, and z-scores can be used to find the cumulative area to the left of a given value, which is crucial for determining percentiles.
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Finding Standard Normal Probabilities using z-Table

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 be less than or equal to a specific z-score. This concept is vital for interpreting z-scores in terms of percentiles, allowing for the assessment of how a particular score compares to the overall distribution.
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Probabilities Between Two Values
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