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Magazine Chapter 6: Problem 56
Determine the value of \(z\) so that the area under the standard normal curve a. in the right tail is 0250 b. in the left tail is \(.0500\) c. in the left tail is \(.0010\) d. in the right tail is. 0100
Short Answer
Expert verified
The solutions are \(z = 1.96, z = -1.64, z = -3.09, z = 2.32\) for parts a, b, c, and d respectively.
Step by step solution
01
Understand the Problem
We are given the area under the normal curve in the left or right tail, and we are asked to determine the corresponding value of \(z\), which is the standard deviation units. We know that a table of Z scores can help us in solving this problem.
02
Use the Z Score Table
We know that for a standard normal curve, the total area under the curve is 1. So, to solve these questions, you'll just subtract the given area from 1 if it's in the right tail and look for the remainder in the Z score table. Find the Z score that corresponds to the calculated entry on the standard normal table. In case the area is in the left tail, there is no need for subtraction from 1.
03
Solve Part a
The area in the right tail is 0.0250. The remainder after subtraction from 1 is \(1 - 0.0250 = 0.9750\). The Z value corresponding to 0.9750 in the Z table is approximately \(1.96\). So, \(z = 1.96\) for the area in the right tail being 0.0250.
04
Solve Part b
The area in the left tail is \(0.0500\). So, the corresponding value of \(z\) in the Z table is approximately \(-1.64\). Thus, \(z = -1.64\) for the area in the left tail being \(0.0500\).
05
Solve Part c
The area in the left tail is \(0.0010\). The corresponding \(z\) value in the Z table is \(-3.09\). Thus, \(z = -3.09\) for the area in the left tail being \(0.0010\).
06
Solve Part d
The area in the right tail is \(0.0100\). After subtracting this from 1, we get \(0.9900\). The corresponding \(z\) value in the Z table is approximately \(2.32\). So, \(z = 2.32\) for the area in the right tail being \(0.0100\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the Z-Score
The concept of a "z-score" is fundamental in statistics, particularly when dealing with the normal distribution. A z-score measures how many standard deviations an element is from the mean of the data set.
Understanding this allows one to compare data across different normal distributions, which is useful in hypothesis testing and other analyses.
If a z-score is 0, it means the data point's score is identical to the mean score. If a z-score is positive, the data point is above the mean; if negative, below the mean.
Understanding this allows one to compare data across different normal distributions, which is useful in hypothesis testing and other analyses.
If a z-score is 0, it means the data point's score is identical to the mean score. If a z-score is positive, the data point is above the mean; if negative, below the mean.
- Calculated as: \( z = \frac{(X - \mu)}{\sigma} \), where \( X \) is the value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
- Helps to determine the likelihood of a score occurring within a normal distribution.
- Important for converting raw data into a standard form to calculate probabilities.
Exploring the Right Tail Area
In statistical analysis using the normal distribution, the "right tail area" refers to the area under the curve to the right of a specified z-score.
The area indicates the probability of observing a value greater than your specified z-score.
The area indicates the probability of observing a value greater than your specified z-score.
- For example, finding a z-score with a right tail area of 0.0250 means there's a 2.5% chance of observing a value larger than the specified z-score.
- To calculate the z-score, one typically finds the complement area by subtracting the tail area from 1 and then using a Z-score table or calculator.
- Used in hypothesis testing and determining significance if a result falls into the critical region on the right tail.
Understanding the Left Tail Area
Similarly, the "left tail area" pertains to the area under the normal curve to the left of a given z-score. This area represents the probability of observing a value less than the specified z-score.
It's important in understanding distributions, particularly skewness towards lower values.
It's important in understanding distributions, particularly skewness towards lower values.
- For example, a left tail area of 0.0500 indicates a 5% likelihood of a score falling below that specific z-score.
- No need for complements when looking up a left tail area, as it directly corresponds to z-score values.
- Vital for certain types of hypothesis testing, where we test if a sample is significantly smaller than the mean.
Using the Normal Curve Table
The "normal curve table," or Z-table, is a critical tool in statistics for finding the area under a standard normal curve associated with a given z-score.
It allows us to understand probabilities associated with normal distributions, which are symmetrical around the mean.
It allows us to understand probabilities associated with normal distributions, which are symmetrical around the mean.
- The table lists z-scores along with their corresponding tail areas or cumulative probabilities.
- Z-tables can show either left or right tail probabilities, adapt your lookup based on the tail's direction.
- Players rely on it for quick reference and transforming distributions into a common scale.
- Understanding how to read the table is key for tasks such as setting confidence intervals and conducting hypothesis tests.
