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Chapter 6: Problem 31

Let \(z\) be a standard normal random variable with mean \(\mu=0\) and standard deviation \(\sigma=1 .\) Find the value \(c\) that satisfies the inequalities given. $$ P(-c

Short Answer

Expert verified
Answer: The value of "c" is 1.37.

Step by step solution

01

Find the Z-score corresponding to the area 0.9131

Using the standard normal distribution table (Z-table), find the Z-score that corresponds to an area (or probability) of 0.9131. Look up the Z-table and find the closest value to 0.9131. The closest value is 0.9130, and its corresponding Z-score is 1.37.
02

Determine the value of "c"

Since the Z-score corresponding to an area of 0.9131 is 1.37, we know that c = 1.37 because the inequality $$P(-c<z<c)=.8262$$ ensures that the area under the curve between -c and c corresponds to 0.8262. So, the value of c is 1.37.

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Key Concepts

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

Z-score
Understanding the Z-score is crucial when dealing with standard normal distributions. A Z-score, also known as a standard score, quantifies the number of standard deviations a given data point is from the mean. In a standard normal distribution, where the mean \(\mu=0\) and the standard deviation \(\sigma=1\), the Z-score for a value is the same as that value itself.

For instance, if we're looking to find a value \(c\) given the probability \(P(-c
Probability
Probability is a measure of the likelihood that a certain event will occur. In terms of the standard normal distribution, the area under the curve within specified bounds represents the probability of a random variable falling within that range. The integral of the normal density function over a range \(a\) to \(b\) gives us this probability.

In the context of our exercise, the probability \(P(-c
Normal Random Variable
A normal random variable is a variable that follows a normal distribution, which is a continuous probability distribution characterized by its bell-shaped curve, known as the Gaussian function. When a normal random variable has a mean of zero and a standard deviation of one, it is referred to as a standard normal random variable or Z-variable.

The reason this is so useful in statistics and probability theory is because of the standard normal distribution's property of being a 'standard' measure, allowing us to compare different datasets by bringing them onto the same scale. This is accomplished through the process of standardization, converting values to Z-scores.

Returning to our example, \(z\) is a standard normal random variable, which means we can directly apply the Z-table to find probabilities for associated ranges without adjusting for mean and standard deviation. This simplifies the computation and makes the concept of Z-scores broadly applicable.

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Most popular questions from this chapter

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