### Understanding Binomial Distribution#### Problem StatementAbout 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 Solve1. **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]### Explanation1. **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.### ApplicationGiven:- $ n = 110 $- $ p = 0.40 $ (from the first input box after converting 40% to a decimal)#### Example Calculation1. **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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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. Convert the percentage of the population to a decimal: p: Compute the mean and standard of this size sample of this binomial distribution: Mean: Standard Deviation:

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. Convert the percentage of the population to a decimal: p: Compute the mean and standard of this size sample of this binomial distribution: Mean: Standard Deviation:

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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Concept explainers

Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.

Question

### 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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