The weight of a certain type of brick has an expectation of 1.12 kilograms with a variance of 0.0009 kilograms². How many bricks would need to be selected so that the average weight has a standard deviation of no more than 0.002 kilograms?
The weight of a certain type of brick has an expectation of 1.12 kilograms with a variance of 0.0009 kilograms². How many bricks would need to be selected so that the average weight has a standard deviation of no more than 0.002 kilograms?
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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Problem #24
![### Problem Statement
The weight of a certain type of brick has the following characteristics:
- **Expectation (Mean Weight)**: 1.12 kilograms
- **Variance**: 0.0009 kilograms²
**Question:**
How many bricks would need to be selected so that the average weight has a standard deviation of no more than 0.002 kilograms?
### Calculation
To find the number of bricks required, we use the relationship between the standard deviation of a sample mean and the number of samples. The formula is:
\[ \text{SD}(\bar{X}) = \frac{\sigma}{\sqrt{n}} \]
Where:
- \(\text{SD}(\bar{X})\) is the standard deviation of the sample mean.
- \(\sigma\) is the standard deviation of the population.
- \(n\) is the number of samples.
Given:
- Desired standard deviation of the average weight: 0.002 kilograms
- Population variance: 0.0009 kilograms²
First, calculate the population standard deviation (\(\sigma\)):
\[ \sigma = \sqrt{0.0009} = 0.03 \text{ kilograms} \]
Now plug the values into the formula and solve for \(n\):
\[ 0.002 = \frac{0.03}{\sqrt{n}} \]
\[ \sqrt{n} = \frac{0.03}{0.002} \]
\[ \sqrt{n} = 15 \]
\[ n = 15^2 \]
\[ n = 225 \]
### Answer
**225 bricks** would need to be selected to ensure the average weight has a standard deviation of no more than 0.002 kilograms.
### Visual Explanation
**Graphical Component:**
There are no graphs or diagrams provided in the original problem statement image. The image primarily consists of the textual problem and a bar, possibly meant for additional interactive elements or emphasis, which includes:
- A blue section with an information icon
- An orange section with an exclamation mark icon
These visual features are not essential to the understanding or solving of the problem but may be used for interactive or highlighting purposes on the educational website.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2d1efc14-51aa-4332-a70c-7292f27466c5%2F7959e104-5478-47dd-b3a7-c3b5a83a24f5%2Fuyc8rbf_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
The weight of a certain type of brick has the following characteristics:
- **Expectation (Mean Weight)**: 1.12 kilograms
- **Variance**: 0.0009 kilograms²
**Question:**
How many bricks would need to be selected so that the average weight has a standard deviation of no more than 0.002 kilograms?
### Calculation
To find the number of bricks required, we use the relationship between the standard deviation of a sample mean and the number of samples. The formula is:
\[ \text{SD}(\bar{X}) = \frac{\sigma}{\sqrt{n}} \]
Where:
- \(\text{SD}(\bar{X})\) is the standard deviation of the sample mean.
- \(\sigma\) is the standard deviation of the population.
- \(n\) is the number of samples.
Given:
- Desired standard deviation of the average weight: 0.002 kilograms
- Population variance: 0.0009 kilograms²
First, calculate the population standard deviation (\(\sigma\)):
\[ \sigma = \sqrt{0.0009} = 0.03 \text{ kilograms} \]
Now plug the values into the formula and solve for \(n\):
\[ 0.002 = \frac{0.03}{\sqrt{n}} \]
\[ \sqrt{n} = \frac{0.03}{0.002} \]
\[ \sqrt{n} = 15 \]
\[ n = 15^2 \]
\[ n = 225 \]
### Answer
**225 bricks** would need to be selected to ensure the average weight has a standard deviation of no more than 0.002 kilograms.
### Visual Explanation
**Graphical Component:**
There are no graphs or diagrams provided in the original problem statement image. The image primarily consists of the textual problem and a bar, possibly meant for additional interactive elements or emphasis, which includes:
- A blue section with an information icon
- An orange section with an exclamation mark icon
These visual features are not essential to the understanding or solving of the problem but may be used for interactive or highlighting purposes on the educational website.
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