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Magazine Chapter 22: Problem 26
Solve the given problems. With \(21 \%\) of the area under the normal curve between \(z_{1}\) and \(z_{2}\) to the right of \(z_{1}=0.8,\) find \(z_{2}\)
Short Answer
Expert verified
The value of \( z_2 \) is approximately 2.88.
Step by step solution
01
Understand the Problem Statement
We need to find the value of \( z_2 \) such that the area under a normal curve between \( z_1 = 0.8 \) and \( z_2 \) is \( 0.21 \). The area to the right of \( z_1 \) starts at \( 0.8 \). We are specifically looking for the \( z \) value where the total area between \( z_1 \) and \( z_2 \) equals \( 0.21 \).
02
Calculate Initial Areas Using z-Tables
From standard normal distribution tables, find the area to the left of \( z_1 = 0.8 \). This area represents the cumulative probability up to \( z_1 = 0.8 \). The area to the left is approximately \( 0.7881 \). Therefore, the area to the right of \( z_1 = 0.8 \) is \( 1 - 0.7881 = 0.2119 \).
03
Determine Area to the Left of z2
Since the total area under the curve is 1, and assuming \( z_2 \) is to the right of \( z_1 \), the area between \( z_1 \) and \( z_2 \) is 0.21. Hence, the area to the left of \( z_2 \) is \( 0.7881 + 0.21 = 0.9981 \).
04
Find z2 Using z-Tables
Check the standard normal distribution tables to find the \( z \) value corresponding to a cumulative probability closest to \( 0.9981 \). This \( z \) value is our \( z_2 \). The closest value in the z-table for \( 0.9981 \) corresponds to a \( z \)-value of approximately \( 2.88 \).
05
Final Step: Verify the Result
Verify that using a \( z_2 \) of \( 2.88 \), the area beyond \( z_1 = 0.8 \) up to \( z_2 = 2.88 \) results in approximately \( 0.21 \). Double-check with the z-table to ensure \( 0.9981 - 0.7881 = 0.21 \) holds.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
z-values
In the realm of statistics, z-values play a critical role when working with data that follows a normal distribution. A z-value, often referred to as a z-score, measures how many standard deviations an element is from the mean.
A positive z-value indicates the score is above the mean, while a negative value means it's below the mean. This is crucial because it standardizes different datasets, allowing us to compare them on the same scale.
A positive z-value indicates the score is above the mean, while a negative value means it's below the mean. This is crucial because it standardizes different datasets, allowing us to compare them on the same scale.
- For example, a z-value of 0.8 tells us that 0.8 standard deviations separate the value from the mean of the data set.
- In our exercise, we have a starting point at a z-value of 0.8, and we are tasked with finding another z-value, called \( z_2 \), which together with the initial point encloses a specific area under the curve.
cumulative probability
When talking about cumulative probability, we refer to the probability that a random variable will take a value less than or equal to a certain threshold. In terms of a normal distribution, it's the total area under the curve to the left of a particular z-value.
- The cumulative probability gives an indication of how much percentage of data falls below a certain z-score.
- In our problem, the cumulative probability for \( z_1 = 0.8 \) is found to be approximately 0.7881. This means that 78.81% of the data lies below a z-score of 0.8.
standard normal distribution
The concept of the standard normal distribution is foundational in statistics. It represents a normal distribution with a mean of 0 and a standard deviation of 1.
When any set of data is standardized using z-scores, it converts into this unified form, making it comparably easy to interpret through z-tables or similar statistical tools.
When any set of data is standardized using z-scores, it converts into this unified form, making it comparably easy to interpret through z-tables or similar statistical tools.
- This transformation is paramount when working on exercises requiring a comparison or determination of areas under normal curves, as in our specified task.
- By using the standard normal distribution, we could easily find cumulative probabilities and subsequently determine the area between any two z-values.
area under curve
The area under the curve, especially in the context of normal distribution, signifies total probability. Every point under this bell-shaped curve adds up to 1, or 100%.
Understanding this area is key for solving problems related to probabilities and z-scores.
Understanding this area is key for solving problems related to probabilities and z-scores.
- For instance, when asked to find the region between two points, you're essentially calculating this area.
- In our exercise, we know that the area between two z-values, \( z_1 \) and \( z_2 \), is 0.21, contributing to our understanding of where \( z_2 \) must be.
