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Suppose $X \sim N(9,3) .$ What is the $z$ -score of $x=9 ?$

   Suppose $X \sim N(9,3) .$ What is the $z$ -score of $x=9 ?$

Introductory Statistics

Introductory Statistics

Barbara Illowsky,… 1st Edition

Chapter 6, Problem 24

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Solved by Expert Bryan Meares

02/21/2020

Step 1

It is calculated using the formula: \[ Z = \frac{(X - \mu)}{\sigma} \] where: - \(X\) is the element - \(\mu\) is the mean - \(\sigma\) is the standard deviation Show more…

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Suppose $X \sim N(9,3) .$ What is the $z$ -score of $x=9 ?$

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Key Concepts

Z-Score

A z-score represents the number of standard deviations a data point is from the mean of its distribution. It is calculated using the formula z = (x - mean) / standard deviation. The z-score is a key concept in statistics for identifying how unusual a particular observation is within the context of its distribution.

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric around its mean. It is characterized by its bell-shaped curve and is defined by two parameters: the mean, which centers the distribution, and the variance (or standard deviation), which determines the spread. Understanding the normal distribution is fundamental in statistics because many variables tend to be normally distributed in natural and social phenomena.

Standardization

Standardization is the process of converting a normal random variable into a standard normal variable. This is done by subtracting the mean from the value of interest and then dividing by the standard deviation. This transformation allows comparisons between values from different normal distributions and facilitates the use of standard normal tables or computational tools for finding probabilities and percentiles.

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