Finding a z-Score Given an Area In Exercises 23–30, find the indicated z-score.


Find the positive z-score for which 12% of the distribution’s area lies between and z.

Verified step by step guidance

1. Step 1: Understand the problem. You are tasked with finding the positive z-score such that 12% (or 0.12) of the area under the standard normal distribution curve lies between 0 and the z-score. This means the area to the left of the z-score is 0.5 (from the left of the mean to 0) plus 0.12.
2. Step 2: Calculate the cumulative area to the left of the z-score. Add the area from the mean to the z-score (0.12) to the area from the far left of the curve to the mean (0.5). This gives a cumulative area of 0.5 + 0.12 = 0.62.
3. Step 3: Use a z-table or statistical software to find the z-score corresponding to a cumulative area of 0.62. A z-table provides the cumulative probability for a given z-score, so you will look for the value closest to 0.62 in the table.
4. Step 4: Identify the z-score from the z-table or software. Locate the row and column in the z-table that correspond to the cumulative probability of 0.62. The intersection of the row and column gives the z-score.
5. Step 5: Verify your result. Ensure that the z-score you found is positive and that the cumulative area to the left of this z-score matches 0.62. This confirms that 12% of the area lies between the mean and the z-score.

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 and for comparing data points from different distributions.

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 z-table, which provides the area (probability) to the left of a given z-score. Understanding this distribution is crucial for finding probabilities and z-scores, as it allows for the conversion of any normal distribution into a standard form for easier analysis.

Area Under the Curve

In statistics, the area under the curve (AUC) of a probability distribution represents the likelihood of a random variable falling within a particular range. For the standard normal distribution, this area can be used to find probabilities associated with z-scores. In the context of the question, finding the z-score for which a specific area (12% in this case) lies between the mean and that z-score involves understanding how to interpret and calculate areas under the normal curve.