MATLAB: An Introduction with Applications
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ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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![### Understanding Binomial Distribution
#### Problem Statement
About 40% of a population are of a particular ethnic group. 110 people are randomly selected from this population. Round all answers to 3 decimal places.
#### Steps to Solve
1. **Convert the percentage of the population to a decimal:**
\( p: \) [Input Box]
2. **Compute the mean and standard deviation of this size sample of this binomial distribution:**
**Mean:**
\(\mu: \) [Input Box]
**Standard Deviation:**
\(\sigma: \) [Input Box]
### Explanation
1. **Convert Percentage to Decimal:**
To convert a percentage to a decimal, divide the percentage by 100.
Formula: \( \text{Decimal Value} = \frac{\text{Percentage}}{100} \)
2. **Computing the Mean of a Binomial Distribution:**
The mean (\(\mu\)) of a binomial distribution is calculated using the formula:
\[
\mu = n \times p
\]
where \( n \) is the number of trials and \( p \) is the probability of success.
3. **Computing the Standard Deviation of a Binomial Distribution:**
The standard deviation (\(\sigma\)) of a binomial distribution is calculated using the formula:
\[
\sigma = \sqrt{n \times p \times (1 - p)}
\]
This measures the dispersion of the sample proportions.
### Application
Given:
- \( n = 110 \)
- \( p = 0.40 \) (from the first input box after converting 40% to a decimal)
#### Example Calculation
1. **Decimal Conversion:**
If initially given 40%, converting this to a decimal:
\[
p = \frac{40}{100} = 0.400
\]
2. **Mean Calculation:**
\[
\mu = 110 \times 0.40 = 44.000
\]
3. **Standard Deviation Calculation:**
\[
\sigma = \sqrt{110 \times 0.40 \times (1 - 0.40)}
\]
\[
\sigma = \sqrt{110 \times 0.40 \times 0.60}
\](https://content.bartleby.com/qna-images/question/1a3586c7-75e0-478e-8b93-fa570f1e3b86/b2ba91f1-3f5e-4901-9ae9-dc6d653349ad/rpvb5v_processed.png)
Transcribed Image Text:### Understanding Binomial Distribution
#### Problem Statement
About 40% of a population are of a particular ethnic group. 110 people are randomly selected from this population. Round all answers to 3 decimal places.
#### Steps to Solve
1. **Convert the percentage of the population to a decimal:**
\( p: \) [Input Box]
2. **Compute the mean and standard deviation of this size sample of this binomial distribution:**
**Mean:**
\(\mu: \) [Input Box]
**Standard Deviation:**
\(\sigma: \) [Input Box]
### Explanation
1. **Convert Percentage to Decimal:**
To convert a percentage to a decimal, divide the percentage by 100.
Formula: \( \text{Decimal Value} = \frac{\text{Percentage}}{100} \)
2. **Computing the Mean of a Binomial Distribution:**
The mean (\(\mu\)) of a binomial distribution is calculated using the formula:
\[
\mu = n \times p
\]
where \( n \) is the number of trials and \( p \) is the probability of success.
3. **Computing the Standard Deviation of a Binomial Distribution:**
The standard deviation (\(\sigma\)) of a binomial distribution is calculated using the formula:
\[
\sigma = \sqrt{n \times p \times (1 - p)}
\]
This measures the dispersion of the sample proportions.
### Application
Given:
- \( n = 110 \)
- \( p = 0.40 \) (from the first input box after converting 40% to a decimal)
#### Example Calculation
1. **Decimal Conversion:**
If initially given 40%, converting this to a decimal:
\[
p = \frac{40}{100} = 0.400
\]
2. **Mean Calculation:**
\[
\mu = 110 \times 0.40 = 44.000
\]
3. **Standard Deviation Calculation:**
\[
\sigma = \sqrt{110 \times 0.40 \times (1 - 0.40)}
\]
\[
\sigma = \sqrt{110 \times 0.40 \times 0.60}
\
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