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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
\(z_1 \approx 1.4\)

Step by step solution

01

Understand the Given Information

We know that 5.8% of the area under the normal curve lies between \(z_1\) and \(z_2 = 2.0\) to the left of \(z_2 = 2.0\). This implies that the area from \(z = -\infty\) to \(z = 2.0\) minus the area from \(z = -\infty\) to \(z = z_1\) equals 5.8%. We need to use this information to find \(z_1\).
02

Convert Percentage to Decimal Format

Convert the percentage of the area between \(z_1\) and \(z_2\) to decimal form by dividing by 100. Thus, \(5.8\% = 0.058\) as a decimal.
03

Determine Area to the Left of \(z_2\)

Use the standard normal distribution table or a calculator to find the area under the normal curve to the left of \(z_2 = 2.0\). From standard normal distribution tables, the cumulative area to the left of \(z = 2.0\) is approximately 0.9772.
04

Find Area to the Left of \(z_1\)

Since the area between \(z_1\) and \(z_2 = 2.0\) is 0.058, subtract this from the cumulative area up to \(z = 2.0\) to find the area to the left of \(z_1\):\[ \text{Area to the left of } z_1 = 0.9772 - 0.058 \approx 0.9192. \]
05

Find \(z_1\) using Cumulative Area

Using the standard normal distribution table or calculator, find the \(z\)-value that corresponds to the cumulative area of 0.9192. By referencing the table, this area corresponds to approximately \(z_1 \approx 1.4\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

z-score
The concept of a z-score is central to understanding normal distribution problems. A z-score tells us how many standard deviations an element is from the mean of a dataset.
A positive z-score means the data point is above the mean, while a negative z-score indicates it is below the mean.
  • The formula for calculating a z-score is: \( z = \frac{X - \mu}{\sigma} \)
  • \( X \) is the data point, \( \mu \) is the mean, and \( \sigma \) is the standard deviation of the dataset.
In the context of the exercise, \( z_2 = 2.0 \) represents a point that is 2 standard deviations away from the mean in the positive direction. This tells us where \( z_2 \) is located on the normal distribution curve and allows us to find cumulative areas associated with this z-score using standard normal tables.
cumulative area
A cumulative area under a standard normal distribution curve represents the probability that a z-score is less than or equal to a specific value.
Understanding cumulative areas is crucial because they allow us to infer probabilities and find z-scores within a normal distribution.
  • Cumulative areas grow as you move left to right along the z-axis on the curve.
  • The total area under the curve is always 1, or 100%.
In the exercise, the cumulative area to the left of \( z = 2.0 \) is 0.9772, meaning that approximately 97.72% of the data lies within two standard deviations above the mean.
To find the area to the left of \( z_1 \), we subtract 0.058 (which is 5.8%) from the area to the left of \( z_2 = 2.0 \), resulting in a new cumulative area of approximately 0.9192.
standard normal table
The standard normal table, also known as the Z-table, is a mathematical table that provides the cumulative probabilities associated with standard normal z-scores.
These tables make the process of finding probabilities and corresponding z-scores very convenient.
  • The z-table lists z-scores in one column, along with their cumulative probabilities in the adjacent column.
  • It covers z-scores ranging from typically -3.49 to 3.49, which encompasses nearly the entire distribution curve.
In the given exercise, once the cumulative area to the left of \( z_1 \) was determined to be approximately 0.9192, the standard normal table was consulted.
This helped locate the z-score closest to 0.9192, which is approximately \( z_1 = 1.4 \).
Students can use this table to quickly identify z-scores for any known cumulative probability.

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Most popular questions from this chapter

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