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Magazine Chapter 6: Problem 20
Find the area under the standard normal distribution curve. Between z = 0.24 and z = ?1.12
Short Answer
Expert verified
The area between z = 0.24 and z = -1.12 is 0.4634.
Step by step solution
01
Understand the Problem
We are tasked with finding the area under the standard normal distribution curve between two z-scores, z = 0.24 and z = -1.12. The area corresponds to the probability between these scores under the normal distribution.
02
Use Standard Normal Distribution Table
We look up the area associated with each z-score in the standard normal distribution table (z-table). For z = 0.24, the area to the left is approximately 0.5948. For z = -1.12, the area to the left is approximately 0.1314.
03
Calculate the Area Between the Z-scores
To find the area between z = 0.24 and z = -1.12, subtract the smaller area from the larger area. Therefore, we calculate 0.5948 - 0.1314.
04
Perform the Subtraction
Calculate 0.5948 - 0.1314 = 0.4634. Thus, the area between z = 0.24 and z = -1.12 is 0.4634.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Z-Scores
Z-scores are a way of expressing a value's position relative to the mean of a standard normal distribution. When we talk about z-scores, we're essentially talking about how many standard deviations away a particular value is from the mean. In the context of a standard normal distribution, the mean is always 0, and the standard deviation is 1.
To calculate a z-score, use the formula: \[z = \frac{X - \mu}{\sigma}\]Where:
To calculate a z-score, use the formula: \[z = \frac{X - \mu}{\sigma}\]Where:
- \(X\) is the value being measured
- \(\mu\) is the mean of the distribution
- \(\sigma\) is the standard deviation
Probability in Normal Distribution
Probability is a measure of the chance that a certain event will occur. In the context of the standard normal distribution, probabilities can be visualized as the area under the curve.
For a particular z-score, the probability refers to the proportion of the data that falls to the left of that z-score within the distribution. Thus, if we want to know the probability of a value being below a certain point, we look at the area corresponding to that z-score.
When dealing with multiple z-scores, finding the probability of a value being between two scores involves calculating the area between these scores on the standard normal curve.
For a particular z-score, the probability refers to the proportion of the data that falls to the left of that z-score within the distribution. Thus, if we want to know the probability of a value being below a certain point, we look at the area corresponding to that z-score.
When dealing with multiple z-scores, finding the probability of a value being between two scores involves calculating the area between these scores on the standard normal curve.
Using the Z-Table
The z-table is a valuable tool for finding probabilities related to the standard normal distribution. It's a tabulated set of values that tells us the area under the curve to the left of a given z-score.
When using a z-table, each z-score corresponds to a specific area, or probability, under the curve. For example, for a z-score of 0.24, the table might give us 0.5948, indicating that 59.48% of the data falls to the left of this z-score in a standard normal distribution.
When using a z-table, each z-score corresponds to a specific area, or probability, under the curve. For example, for a z-score of 0.24, the table might give us 0.5948, indicating that 59.48% of the data falls to the left of this z-score in a standard normal distribution.
- Find the z-score in the table.
- Read the cumulative area associated with that score.
- Subtract areas to find probabilities between two z-scores if needed.
Area Under the Curve
The area under the curve in a standard normal distribution is vital because it represents probabilities. When we talk about the 'area' under this curve, we mean the total probability that a variable falls within a specific range.
In the standard normal distribution, the entire area under the curve equals 1, or 100%, because the curve encompasses the whole range of probabilities.
To find an area between two z-scores:
In the standard normal distribution, the entire area under the curve equals 1, or 100%, because the curve encompasses the whole range of probabilities.
To find an area between two z-scores:
- Look up each z-score in the z-table to find their corresponding left-tail probabilities.
- Subtract the smaller probabilistic area from the larger one to find the area between the two z-scores.
