Problem 16
Question
Find \(b\) if \(P(Z \geq b)=0.73\), where \(Z\) is a random variable with a standard normal distribution \((\mu=0, \sigma=1)\).
Step-by-Step Solution
Verified Answer
\(b = -0.61\)
1Step 1: Identify the Given Information
We are given the probability that a standard normal random variable is greater than a value, which is: \(P(Z \, \geq \, b) = 0.73\). We need to find the corresponding value of \(b\).
2Step 2: Convert to Cumulative Probability
Standard normal tables typically provide cumulative probabilities from the left, so we need to convert the given probability. Since \(P(Z \, \geq \, b) = 0.73\), it follows that \(P(Z \, < \, b) = 1 - 0.73 = 0.27\).
3Step 3: Use Z-Table to Find the Z-Score
Look up the cumulative probability of 0.27 in the standard normal (Z) table to find the corresponding z-score.
4Step 4: Verify and Determine b
From the Z-table, the z-score that corresponds to a cumulative probability of 0.27 is approximately \(-0.61\). Thus, \(b = -0.61\).
Key Concepts
Cumulative ProbabilityZ-ScoreZ-Table
Cumulative Probability
In statistics, cumulative probability refers to the probability that a random variable takes a value less than or equal to a specific value. This is crucial for interpreting the probabilities in standard normal distribution problems.
For instance, given that the standard normal distribution is symmetric around zero, cumulative probability for a value is the area under the curve to the left of that value.
In this exercise, we initially have been given a probability greater than a certain value: \(P(Z \geq b)=0.73\). To find the corresponding value of \(b\), we need to convert this into the more standard cumulative probability form: \(P(Z
Understanding cumulative probability helps in using the Z-table effectively, as Z-tables typically provide cumulative probabilities from the left (i.e., \(P(Z
For instance, given that the standard normal distribution is symmetric around zero, cumulative probability for a value is the area under the curve to the left of that value.
In this exercise, we initially have been given a probability greater than a certain value: \(P(Z \geq b)=0.73\). To find the corresponding value of \(b\), we need to convert this into the more standard cumulative probability form: \(P(Z
Understanding cumulative probability helps in using the Z-table effectively, as Z-tables typically provide cumulative probabilities from the left (i.e., \(P(Z
Z-Score
A Z-score tells us how many standard deviations a value is from the mean of the distribution.
In the context of a standard normal distribution, a Z-score indicates the position of a value relative to the mean, which is 0, with a standard deviation of 1.
For example, a Z-score of -0.61 means the value is 0.61 standard deviations below the mean.
To find the Z-score corresponding to a specific probability, we use the Z-table. In this problem, we need the Z-score for the cumulative probability of 0.27. By looking up 0.27 in the Z-table, we found the Z-score to be approximately -0.61.
This Z-score helps us determine the critical threshold (in our case, \(b=-0.61\)) for the given probability.
In the context of a standard normal distribution, a Z-score indicates the position of a value relative to the mean, which is 0, with a standard deviation of 1.
For example, a Z-score of -0.61 means the value is 0.61 standard deviations below the mean.
To find the Z-score corresponding to a specific probability, we use the Z-table. In this problem, we need the Z-score for the cumulative probability of 0.27. By looking up 0.27 in the Z-table, we found the Z-score to be approximately -0.61.
This Z-score helps us determine the critical threshold (in our case, \(b=-0.61\)) for the given probability.
Z-Table
A Z-table, also known as the standard normal distribution table, lists the cumulative probability of a standard normal random variable being less than or equal to a given Z-score.
Here's how to use the Z-table:
In this exercise, we sought the cumulative probability of 0.27. By finding this probability in the Z-table, we identified the Z-score at approximately -0.61. This value tells us that 0.27 of the distribution lies to the left of -0.61 on the standard normal curve.
The Z-table is a vital tool in statistics for relating probabilities and Z-scores, which is essential for solving problems involving standard normal distributions like the one in this exercise.
Here's how to use the Z-table:
- Find the cumulative probability in the table that matches your desired probability.
- Locate the corresponding Z-score for this probability.
In this exercise, we sought the cumulative probability of 0.27. By finding this probability in the Z-table, we identified the Z-score at approximately -0.61. This value tells us that 0.27 of the distribution lies to the left of -0.61 on the standard normal curve.
The Z-table is a vital tool in statistics for relating probabilities and Z-scores, which is essential for solving problems involving standard normal distributions like the one in this exercise.
Other exercises in this chapter
Problem 14
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