standard deviation of 1.3 inches. If one item is chosen at random, what is the probability that it is less than 8.2 inches long? > Next Question
standard deviation of 1.3 inches. If one item is chosen at random, what is the probability that it is less than 8.2 inches long? > Next Question
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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![### Content for Educational Website
**Probability and Statistical Distributions**
#### Understanding Normal Distributions with Practical Examples
A manufacturer knows that their items have a normally distributed length, with a mean of 7.3 inches, and standard deviation of 1.3 inches.
**Problem:**
If one item is chosen at random, what is the probability that it is less than 8.2 inches long?
In this exercise, we explore how to calculate probabilities for normally distributed variables. Given the mean (μ) and standard deviation (σ) of a normal distribution, we can determine the likelihood that a randomly chosen item will fall within a certain range.
We can use the Z-score formula to standardize the variable:
\[ Z = \frac{X - \mu}{\sigma} \]
Where:
- \( X \) is the value we are interested in (8.2 inches in this case)
- \( \mu \) is the mean (7.3 inches)
- \( \sigma \) is the standard deviation (1.3 inches)
**Next Step:**
Calculate the Z-score and look up the corresponding probability in the standard normal distribution table, or use a calculator that provides cumulative probabilities for the standard normal distribution.
**Interactive Learning:**
Use the input field to calculate and submit your answer. Then proceed to the next question to deepen your understanding.
\[ \textbf{Next Question} \]
</ Educational Content >
This page also includes an interactive component where students can enter their calculated probabilities and proceed to subsequent questions to test their understanding of normal distributions.
Note: A basic understanding of statistical concepts such as Z-scores and the standard normal distribution table is required to solve this problem efficiently. Also, access to a statistical calculator or software that provides cumulative probabilities for the standard normal distribution might be helpful.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faa28b18f-3e2d-4592-9d09-7b963ee49630%2F2e8459cd-004d-4c1e-b095-b687b29e1fa7%2F3tnry6n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Content for Educational Website
**Probability and Statistical Distributions**
#### Understanding Normal Distributions with Practical Examples
A manufacturer knows that their items have a normally distributed length, with a mean of 7.3 inches, and standard deviation of 1.3 inches.
**Problem:**
If one item is chosen at random, what is the probability that it is less than 8.2 inches long?
In this exercise, we explore how to calculate probabilities for normally distributed variables. Given the mean (μ) and standard deviation (σ) of a normal distribution, we can determine the likelihood that a randomly chosen item will fall within a certain range.
We can use the Z-score formula to standardize the variable:
\[ Z = \frac{X - \mu}{\sigma} \]
Where:
- \( X \) is the value we are interested in (8.2 inches in this case)
- \( \mu \) is the mean (7.3 inches)
- \( \sigma \) is the standard deviation (1.3 inches)
**Next Step:**
Calculate the Z-score and look up the corresponding probability in the standard normal distribution table, or use a calculator that provides cumulative probabilities for the standard normal distribution.
**Interactive Learning:**
Use the input field to calculate and submit your answer. Then proceed to the next question to deepen your understanding.
\[ \textbf{Next Question} \]
</ Educational Content >
This page also includes an interactive component where students can enter their calculated probabilities and proceed to subsequent questions to test their understanding of normal distributions.
Note: A basic understanding of statistical concepts such as Z-scores and the standard normal distribution table is required to solve this problem efficiently. Also, access to a statistical calculator or software that provides cumulative probabilities for the standard normal distribution might be helpful.
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