Textbook Question
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.
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6. Normal Distribution and Continuous Random Variables
Probabilities & Z-Scores w/ Graphing Calculator
1:35 minutesProblem 5.3.27
Textbook Question
Finding a z-Score Given an Area In Exercises 23–30, find the indicated z-score.
Find the z-score that has 2.275% of the distribution’s area to its left.
Verified step by step guidance1
Step 1: Understand the problem. The z-score represents the number of standard deviations a data point is from the mean in a standard normal distribution. Here, we are tasked with finding the z-score such that 2.275% (or 0.02275 as a decimal) of the distribution's area lies to its left.
Step 2: Convert the given percentage to a cumulative probability. Since the area to the left of the z-score is given as 2.275%, we interpret this as the cumulative probability P(Z < z) = 0.02275.
Step 3: Use a z-table or statistical software to find the z-score corresponding to the cumulative probability of 0.02275. In a z-table, locate the value closest to 0.02275 in the body of the table and identify the corresponding z-score from the row and column headers.
Step 4: If using statistical software or a calculator, use the inverse cumulative distribution function (often denoted as invNorm or similar) to find the z-score. Input the cumulative probability of 0.02275 to obtain the z-score.
Step 5: Interpret the result. The z-score you find will be negative because 2.275% of the area is in the left tail of the standard normal distribution, which 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 and for comparing data points from different distributions.
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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-distribution, which allows for the calculation of probabilities and z-scores. The area under the curve of the standard normal distribution corresponds to probabilities, making it a fundamental tool in statistics for determining how likely a particular z-score is.
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Area Under the Curve
In the context of the normal distribution, the area under the curve represents the probability of a random variable falling within a particular range. For z-scores, this area can be found using z-tables or statistical software. When given a specific area, such as 2.275%, one can determine the corresponding z-score that marks that percentile in the distribution, which is crucial for hypothesis testing and confidence intervals.
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