A Manufacturer knows that their items have a normally distributed length, with a mean of 15.1 inches, and standard deviation of 4.4 inches. If one item is chosen at random, what is the probability that it is less than 14.6 inches long? ( Give answer to 4 decimal places)
A Manufacturer knows that their items have a normally distributed length, with a mean of 15.1 inches, and standard deviation of 4.4 inches. If one item is chosen at random, what is the probability that it is less than 14.6 inches long? ( Give answer to 4 decimal places)
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
![### Understanding Normal Distributions: A Practical Example
In this example, we will examine a scenario where a manufacturer knows that the length of their items follows a normal distribution.
**Scenario Description:**
- **Mean (μ)**: 15.1 inches
- **Standard Deviation (σ)**: 4.4 inches
The manufacturer wants to know the probability that a randomly chosen item is shorter than 14.6 inches.
**Question:**
If one item is chosen at random, what is the probability that it is less than 14.6 inches long? (Provide your answer to four decimal places).
### Step-by-Step Solution:
1. **Identify the Z-Score:**
To find the probability, we first need to convert the length (14.6 inches) to a Z-score.
The formula for the Z-score is:
\[
Z = \frac{X - \mu}{\sigma}
\]
Where:
- \(X\) is the value for which we are finding the Z-score (14.6 inches).
- \(\mu\) is the mean (15.1 inches).
- \(\sigma\) is the standard deviation (4.4 inches).
2. **Calculate the Z-Score:**
\[
Z = \frac{14.6 - 15.1}{4.4} = \frac{-0.5}{4.4} \approx -0.1136
\]
3. **Find the Probability:**
Using a standard normal distribution table or a calculator, we find the probability corresponding to the Z-score of -0.1136.
The probability (P) that a Z-score will be less than -0.1136 is approximately 0.4549.
### Conclusion:
Thus, the probability that a randomly chosen item is less than 14.6 inches long is **0.4549**.
This example illustrates how understanding the properties of the normal distribution allows you to calculate probabilities for different values effectively.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd5176b0c-078c-4393-8a73-67cd4f62d79f%2F5945e062-53cb-4c41-a88d-f96f1637eec1%2F8v0mfu_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Normal Distributions: A Practical Example
In this example, we will examine a scenario where a manufacturer knows that the length of their items follows a normal distribution.
**Scenario Description:**
- **Mean (μ)**: 15.1 inches
- **Standard Deviation (σ)**: 4.4 inches
The manufacturer wants to know the probability that a randomly chosen item is shorter than 14.6 inches.
**Question:**
If one item is chosen at random, what is the probability that it is less than 14.6 inches long? (Provide your answer to four decimal places).
### Step-by-Step Solution:
1. **Identify the Z-Score:**
To find the probability, we first need to convert the length (14.6 inches) to a Z-score.
The formula for the Z-score is:
\[
Z = \frac{X - \mu}{\sigma}
\]
Where:
- \(X\) is the value for which we are finding the Z-score (14.6 inches).
- \(\mu\) is the mean (15.1 inches).
- \(\sigma\) is the standard deviation (4.4 inches).
2. **Calculate the Z-Score:**
\[
Z = \frac{14.6 - 15.1}{4.4} = \frac{-0.5}{4.4} \approx -0.1136
\]
3. **Find the Probability:**
Using a standard normal distribution table or a calculator, we find the probability corresponding to the Z-score of -0.1136.
The probability (P) that a Z-score will be less than -0.1136 is approximately 0.4549.
### Conclusion:
Thus, the probability that a randomly chosen item is less than 14.6 inches long is **0.4549**.
This example illustrates how understanding the properties of the normal distribution allows you to calculate probabilities for different values effectively.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 1 images
Recommended textbooks for you
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman