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Chapter 6: Problem 9

For the standard normal distribution, what does \(z\) represent?

Short Answer

Expert verified
In a standard normal distribution, \(z\) represents a z-score, showing how many standard deviations an element is from the mean.

Step by step solution

01

Define Standard Normal Distribution

A standard normal distribution is a type of distribution in statistics which has a mean of 0 and a standard deviation of 1.
02

Define Z in this context

In the context of a standard normal distribution, \(z\) refers to a z-score. A z-score shows how many standard deviations an element is from the mean.
03

Explain the significance of Z

The value of \(z\) is used to calculate the probability that a score will fall within a certain range. For example, if \(z=1.5\), it means that the value is 1.5 standard deviations from the mean. This is used to determine the likelihood that a future data point will fall within a specified range

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Key Concepts

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

z-score
A z-score is a statistical measure that describes a data point's relationship to the mean of a group of points. In the context of the standard normal distribution, the mean is always 0, and the standard deviation is 1. The z-score helps to determine how many standard deviations an element is from the mean. For instance, a z-score of 2 signifies that the value is 2 standard deviations above the mean, while a z-score of -1 indicates the value is 1 standard deviation below the mean.
Z-scores are essential in standardizing different datasets, making it easier to compare scores on different scales by translating them into a common scale.
mean
In statistics, the mean, or average, is the sum of all values in a dataset divided by the number of values. It provides a central location for the distribution. For a standard normal distribution, the mean is set and assumed to be 0. This makes the z-score calculations straightforward, as they are directly related to this centralized value.
Having a mean of 0 simplifies the interpretation of data in the normal distribution since it balances the values on either side of zero, making the interpretation of deviations (z-scores) clear and simple.
standard deviation
Standard deviation is a measure of the amount of variation or dispersion in a set of values. It tells us how much the individual data points typically differ from the mean. In a standard normal distribution, the standard deviation is always 1.
This uniform standard deviation simplifies calculations and interpretations. For example, when calculating z-scores, each score's distance from the mean is measured in standard deviation units, permitting direct comparison to other datasets also expressed in z-scores.
probability
Probability measures the likelihood of an event occurring. In the context of a standard normal distribution, you can use z-scores to determine the probability that a random variable will fall within a particular range of values.
For example, by using a z-table or a statistical software, you can find the probability that a score is less than or greater than a given z-score. This makes probability a powerful tool in inferential statistics, allowing predictions about population behavior based on sample data. Understanding probabilities through z-scores helps in decision-making processes in various fields, such as finance and risk assessment.

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Most popular questions from this chapter

For a binomial probability distribution, \(n=120\) and \(p=.60\). Let \(x\) be the number of successes in 120 trials. a. Find the mean and standard deviation of this binomial distribution. b. Find \(P(x \leq 69)\) using the normal approximation. c. Find \(P(67 \leq x \leq 73)\) using the normal approximation.

Obtain the area under the standard normal curve a. to the right of \(z=1.43\) b. to the left of \(z=-1.65\) c. to the right of \(z=-.65\) d. to the left of \(z=.89\)

What is the difference between the probability distribution of a discrete random variable and that of a continuous random variable? Explain.

According to a U.S. Census American Community Survey, \(5.44 \%\) of workers in Portland, Oregon, commute to work on their bicycles. Find the probability that in a sample of 400 workers from Portland, Oregon, the number who commute to work on their bicycles is 23 to 27 .

For a binomial probability distribution, \(n=25\) and \(p=.40\). a. Find the probability \(P(8 \leq x \leq 13)\) by using the table of binomial probabilities (Table I of Appendix B). b. Find the probability \(P(8 \leq x \leq 13)\) by using the normal distribution as an approximation to the binomial distribution. What is the difference between this approximation and the exact probability calculated in part a?
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