. A random variable X is normally distributed with a mean of 25 and a standard deviation of 5. What is the probability that a randomly selected individual will have a score at or less than 22? If a sample of size n = 8 was taken from this population, what is the probability that the sample mean is less an 22?
. A random variable X is normally distributed with a mean of 25 and a standard deviation of 5. What is the probability that a randomly selected individual will have a score at or less than 22? If a sample of size n = 8 was taken from this population, what is the probability that the sample mean is less an 22?
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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![**Sample Mean:**
\[ \bar{X} = \frac{\Sigma X}{n} \]
Where:
- \(\Sigma\) = sum (or add up)
- \(\bar{X}\) = sample mean
- \(n\) = total number of scores, or the sample size
For a **population**, mean is represented by the Greek letter mu (\(\mu\)).
---
**Population Variance:**
\[ \sigma^2 = \frac{\Sigma (X - \mu)^2}{N} \]
**Sample Variance:**
\[ s^2 = \frac{\Sigma (X - \bar{X})^2}{n - 1} \]
Remember that the **standard deviation** (\(\sigma\) or \(s\)) is the square root (\(\sqrt{ }\)) of the variance.
---
**Z Score (for one score/individual):**
\[ Z = \frac{X - \mu}{\sigma} \]
**Z Score: One Sample Z test:**
\[ Z = \frac{\bar{X} - \mu}{\sigma / \sqrt{n}} \]
Definitions:
- \(\bar{X}\) = mean of sample
- \(\mu\) = mean of population
- \(\sigma\) = standard deviation of population
- \(n\) = sample size
- \(\sqrt{n}\) = square root of sample size
---
This content is prepared to provide concise formulas and definitions essential for understanding basic statistical calculations and hypothesis testing using the Z score.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7dee35c8-07bc-4f6f-8047-1c0be93c606e%2F0bacce06-c139-4205-b494-a59096dc3399%2Fpahf9hr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Sample Mean:**
\[ \bar{X} = \frac{\Sigma X}{n} \]
Where:
- \(\Sigma\) = sum (or add up)
- \(\bar{X}\) = sample mean
- \(n\) = total number of scores, or the sample size
For a **population**, mean is represented by the Greek letter mu (\(\mu\)).
---
**Population Variance:**
\[ \sigma^2 = \frac{\Sigma (X - \mu)^2}{N} \]
**Sample Variance:**
\[ s^2 = \frac{\Sigma (X - \bar{X})^2}{n - 1} \]
Remember that the **standard deviation** (\(\sigma\) or \(s\)) is the square root (\(\sqrt{ }\)) of the variance.
---
**Z Score (for one score/individual):**
\[ Z = \frac{X - \mu}{\sigma} \]
**Z Score: One Sample Z test:**
\[ Z = \frac{\bar{X} - \mu}{\sigma / \sqrt{n}} \]
Definitions:
- \(\bar{X}\) = mean of sample
- \(\mu\) = mean of population
- \(\sigma\) = standard deviation of population
- \(n\) = sample size
- \(\sqrt{n}\) = square root of sample size
---
This content is prepared to provide concise formulas and definitions essential for understanding basic statistical calculations and hypothesis testing using the Z score.
Transcribed Image Text:**Problem 14: Understanding Normally Distributed Random Variables**
A random variable \( X \) follows a normal distribution with a mean (\( \mu \)) of 25 and a standard deviation (\( \sigma \)) of 5.
**Questions:**
a. *What is the probability that a randomly selected individual will have a score at or less than 22?*
b. *If a sample of size \( n = 8 \) is taken from this population, what is the probability that the sample mean is less than 22?*
---
To solve these problems, you can use the properties of the normal distribution. For part (a), you would calculate the z-score and use the standard normal distribution table to find the probability. For part (b), you would need to consider the sampling distribution of the sample mean and calculate the probability accordingly, often utilizing the Central Limit Theorem.
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