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.

0.6736

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Step 1: Understand the problem. The goal is to find the z-score that corresponds to a cumulative area (or percentile) of 0.6736 under the standard normal distribution curve. The cumulative area represents the probability that a value is less than or equal to the z-score.

Step 2: Recall the properties of the standard normal distribution. The standard normal distribution has a mean (μ) of 0 and a standard deviation (σ) of 1. The cumulative area is the area under the curve to the left of the z-score.

Step 3: Use the Standard Normal Table (also called the z-table) or technology. Locate the cumulative area of 0.6736 in the table. The table provides the cumulative area for various z-scores. Alternatively, use statistical software or a calculator with a built-in function to find the z-score corresponding to this cumulative area.

Step 4: Match the cumulative area to the closest value in the z-table. Identify the row and column that correspond to this cumulative area. The intersection of the row and column gives the z-score. If using technology, input the cumulative area (0.6736) into the appropriate function to directly obtain the z-score.

Step 5: Interpret the result. The z-score represents the number of standard deviations the value is from the mean. A positive z-score indicates the value is above the mean, while a negative z-score indicates it is below the mean.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 symmetrical 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 interpreting statistical data.

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 determining percentiles and understanding how data is distributed relative to the mean.

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