Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 22: Problem 28
Solve the given problems. With \(5.8 \%\) of the area under the normal curve between \(z_{1}\) and \(z_{2}\) to the left of \(z_{2}=2.0,\) find \(z_{1}\)
Short Answer
Expert verified
\(z_1\) is approximately 1.39.
Step by step solution
01
Identify the Problem
We need to find the value of \(z_{1}\) such that the cumulative area between \(z_{1}\) and \(z_{2}\) is equal to \(5.8\%\), with \(z_{2}\) given as \(2.0\).
02
Understanding the Normal Distribution
The total area under the normal distribution curve is 1.0. We're given the area between \(z_{1}\) and \(z_{2}\) as \(5.8\%\) or \(0.058\) in decimal form, and \(z_{2}\) has a cumulative probability of approximately \(0.9772\).
03
Calculate Cumulative Probability for \(z_1\)
To find \(z_{1}\), we subtract \(0.058\) (area between \(z_{1}\) and \(z_{2}\)) from the cumulative probability of \(z_{2}=2.0\), which is \(0.9772\). This gives us \(0.9772 - 0.058 = 0.9192\).
04
Find \(z_1\) Using a Z-table or Calculator
Using a Z-table or a calculator with statistical functions, we find the \(z\)-score that corresponds to a cumulative probability of \(0.9192\). This results in \(z_{1} \approx 1.39\).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Z-score
The z-score is a numerical measurement that describes a value's relation to the mean of a group of values. It's expressed in terms of standard deviations from the mean. In a normal distribution, z-scores allow us to calculate the probability of a value occurring within a certain distribution and compare different data points from different normal distributions.
To find a z-score, use the formula:
\[ z = \frac{(X - \mu)}{\sigma} \]
where \(X\) is the value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. In the context of the problem, the z-scores help us identify where the specified probability area lies on the normal distribution curve.
- A positive z-score indicates the value is above the mean.
- A negative z-score indicates the value is below the mean.
- A z-score of zero indicates the value is exactly at the mean.
To find a z-score, use the formula:
\[ z = \frac{(X - \mu)}{\sigma} \]
where \(X\) is the value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. In the context of the problem, the z-scores help us identify where the specified probability area lies on the normal distribution curve.
Exploring Cumulative Probability
Cumulative probability refers to the probability that a random variable takes on a value less than or equal to a specified value. In the context of a normal distribution, it is the total area under the curve to the left of a given z-score. Understanding this concept is crucial when working with probabilities and normal distributions.
In this exercise, the cumulative probability of \(z_2 = 2.0\) is 0.9772, meaning 97.72% of the data is below this z-score. To find \(z_1\), we subtract the area between \(z_1\) and \(z_2\) (5.8% or 0.058) from 0.9772 to get 0.9192, which represents the cumulative probability of \(z_1\).
- Cumulative probability accumulates as you move from left to right along the x-axis of a normal distribution graph.
- At every point along the curve, the cumulative probability gives the total percentage of values falling below that point.
- When the cumulative probability is close to 1, the value is likely to occur in the upper tail of the distribution.
In this exercise, the cumulative probability of \(z_2 = 2.0\) is 0.9772, meaning 97.72% of the data is below this z-score. To find \(z_1\), we subtract the area between \(z_1\) and \(z_2\) (5.8% or 0.058) from 0.9772 to get 0.9192, which represents the cumulative probability of \(z_1\).
Discovering Area Under the Curve
The area under the curve (AUC) in a normal distribution represents the probability or the likelihood of a random variable falling within a particular range. Each point along the x-axis corresponds to a potential outcome, while the area beneath the curve between two points corresponds to the probability of outcomes falling within that range.
In our problem, the AUC between \(z_1\) and \(z_2\) is given as 0.058. It represents a 5.8% likelihood that a randomly picked value falls within this interval. Knowing the AUC assists in finding the endpoints (\(z_1\) and \(z_2\)) for this probability region.
- The total AUC for a standard normal distribution is always 1, representing 100% probability.
- The distribution is symmetrical around the mean, which is the highest point on the curve.
- By calculating areas under specific segments, we can determine probabilities for events occurring within those segments.
In our problem, the AUC between \(z_1\) and \(z_2\) is given as 0.058. It represents a 5.8% likelihood that a randomly picked value falls within this interval. Knowing the AUC assists in finding the endpoints (\(z_1\) and \(z_2\)) for this probability region.
Navigating the Z-table
A Z-table, also known as a standard normal table, is a mathematical table that shows the percentage of values (cumulative probabilities) to the left of a given z-score in a standard normal distribution. This tool is essential for finding probabilities and percentiles in statistics.
In this exercise, a Z-table was used to determine \(z_1\) by finding the z-score corresponding to the cumulative probability of 0.9192. This process allows precise calculation and interpretation of probabilities in the normal distribution.
- Z-tables typically list z-scores in the left column and along the top row, with cumulative probabilities filling the body of the table.
- Using the Z-table involves locating a specific z-score in the table to find its corresponding cumulative probability.
- Conversely, you can search for a probability in the Z-table to find its associated z-score, as done in the exercise to find \(z_1\).
In this exercise, a Z-table was used to determine \(z_1\) by finding the z-score corresponding to the cumulative probability of 0.9192. This process allows precise calculation and interpretation of probabilities in the normal distribution.
