**Understanding Percentiles in a Normal Distribution**Consider the random variable $ X $, which is normally distributed with a mean ($ \mu $) of 44 and a standard deviation ($ \sigma $) of 9. Our goal is to find the 13th percentile of this distribution.### Key Concepts:- **Normal Distribution**: A probability distribution that is symmetric about the mean, depicting that data near the mean are more frequent in occurrence than data far from the mean.- **Mean ($ \mu $)**: The average or central value.- **Standard Deviation ($ \sigma $)**: A measure of the amount of variation or dispersion in a set of values.- **Percentile**: A measure used in statistics to indicate the value below which a given percentage of observations in a group falls.### Problem Statement:Assume the random variable $ X $ is normally distributed, with mean $ \mu = 44 $ and standard deviation $ \sigma = 9 $. Find the 13th percentile.### Calculate the 13th Percentile:The 13th percentile means that 13% of the data falls below this value. To find this value, you can use statistical tables or software that provides the cumulative distribution function (CDF) of the normal distribution.1. **Z-Score Approach:** - A Z-score is a measure of how many standard deviations an element is from the mean. - Use the standard normal distribution table to find the Z-score corresponding to a cumulative probability of 0.13. - Convert the Z-score to the corresponding $ X $ value using the formula: \[ X = \mu + Z \times \sigma \]2. **Using Statistical Software or Tools:** - Input the mean, standard deviation, and desired percentile to obtain the result directly.By following such methods, you can determine the precise value at the 13th percentile for the given normal distribution.

It is given that mean is 44 and standard deviation is 9.

Answered: Assume the random variable X is…

Assume the random variable X is normally distributed, with mean p= 44 and standard deviation o = 9. Find the 13th percentile.

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.

Question

**Understanding Percentiles in a Normal Distribution**

Consider the random variable $ X $, which is normally distributed with a mean ($ \mu $) of 44 and a standard deviation ($ \sigma $) of 9. Our goal is to find the 13th percentile of this distribution.

### Key Concepts:

- **Normal Distribution**: A probability distribution that is symmetric about the mean, depicting that data near the mean are more frequent in occurrence than data far from the mean.
- **Mean ($ \mu $)**: The average or central value.
- **Standard Deviation ($ \sigma $)**: A measure of the amount of variation or dispersion in a set of values.
- **Percentile**: A measure used in statistics to indicate the value below which a given percentage of observations in a group falls.

### Problem Statement:

Assume the random variable $ X $ is normally distributed, with mean $ \mu = 44 $ and standard deviation $ \sigma = 9 $. Find the 13th percentile.

### Calculate the 13th Percentile:

The 13th percentile means that 13% of the data falls below this value. To find this value, you can use statistical tables or software that provides the cumulative distribution function (CDF) of the normal distribution.

1. **Z-Score Approach:**
- A Z-score is a measure of how many standard deviations an element is from the mean.
- Use the standard normal distribution table to find the Z-score corresponding to a cumulative probability of 0.13.
- Convert the Z-score to the corresponding $ X $ value using the formula:
\[
X = \mu + Z \times \sigma
\]

2. **Using Statistical Software or Tools:**
- Input the mean, standard deviation, and desired percentile to obtain the result directly.

By following such methods, you can determine the precise value at the 13th percentile for the given normal distribution.

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