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Chapter 11: Problem 16

Find \(b\) if \(P(Z \geq b)=0.73\), where \(Z\) is a random variable with a standard normal distribution \((\mu=0, \sigma=1)\).

Short Answer

Expert verified
\(b = -0.61\)

Step by step solution

01

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 \(b\).
02

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 \(P(Z \, < \, b) = 1 - 0.73 = 0.27\).
03

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.
04

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, \(b = -0.61\).

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Key Concepts

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

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
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.
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:
  • 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.

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Most popular questions from this chapter

ACADEMIC TESTING The results of a calculus exam are normally distributed with a mean of \(72.3\) and a standard deviation of \(16.4\). Find the probability that a randomly chosen student's score is between 50 and 75 . If there are 82 students in the class, about how many have scores between 50 and \(75 ?\)

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