A box contains 5 pennies, 3 dimes, 1 quarter, and 1 half-dollar. A coin is drawn at random. Let x be the value of a coin.
A box contains 5 pennies, 3 dimes, 1 quarter, and 1 half-dollar. A coin is drawn at random. Let x be the value of a coin.
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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![### Probability Distribution and Statistical Measures
**Scenario:**
A box contains 5 pennies, 3 dimes, 1 quarter, and 1 half-dollar. A coin is drawn at random. Let \( X \) be the value of a coin.
#### a) Construct a probability distribution for the data.
| \( X \) (Value in cents) | 1 | 10 | 25 | 50 |
|--------------------------|----|----|----|----|
| \( P(X) \) | 0.5| 0.3| 0.1| 0.1|
Values are written as decimals representing the probability of drawing each type of coin.
#### b) Calculate \( E(X) \) (same as \( \mu \)) and variance \( \sigma^2 \), standard deviation \( \sigma \).
- **Expected Value:**
\[
E(X) = \mu = \sum X \cdot P(X) = 3.6
\]
- **Variance \( \sigma^2 \):**
- Calculate using \( \sum (X^2 \cdot P(X)) - \mu^2 = 15.4 - 3.6^2 \)
- **Standard Deviation \( \sigma \):**
- \( \sigma = \sqrt{\text{Variance}} \) (Round to two decimal places)
#### Population Measures
- **Population Mean:** [Answer pending]
- **Population Variance:** [Answer pending]
- **Population Standard Deviation:** [Answer pending]
#### Explanation of Calculations:
1. Each coin type is assigned a probability based on its frequency in the box.
2. The expected value \( E(X) \) is calculated as the sum of products of each value and its probability.
3. Variance is derived by taking each value squared, multiplying by its probability, summing all those values, and subtracting the square of the mean.
4. The standard deviation is the square root of the variance, providing insight into the spread of the coin values around the mean.
This illustration explains how to determine the expected value, variance, and standard deviation of a given random variable from a finite probability distribution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F95e5e52d-4449-4f93-832b-400422add1f0%2F4fb92754-9b35-40de-a9ff-4f984e28d588%2Flzfuwb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Probability Distribution and Statistical Measures
**Scenario:**
A box contains 5 pennies, 3 dimes, 1 quarter, and 1 half-dollar. A coin is drawn at random. Let \( X \) be the value of a coin.
#### a) Construct a probability distribution for the data.
| \( X \) (Value in cents) | 1 | 10 | 25 | 50 |
|--------------------------|----|----|----|----|
| \( P(X) \) | 0.5| 0.3| 0.1| 0.1|
Values are written as decimals representing the probability of drawing each type of coin.
#### b) Calculate \( E(X) \) (same as \( \mu \)) and variance \( \sigma^2 \), standard deviation \( \sigma \).
- **Expected Value:**
\[
E(X) = \mu = \sum X \cdot P(X) = 3.6
\]
- **Variance \( \sigma^2 \):**
- Calculate using \( \sum (X^2 \cdot P(X)) - \mu^2 = 15.4 - 3.6^2 \)
- **Standard Deviation \( \sigma \):**
- \( \sigma = \sqrt{\text{Variance}} \) (Round to two decimal places)
#### Population Measures
- **Population Mean:** [Answer pending]
- **Population Variance:** [Answer pending]
- **Population Standard Deviation:** [Answer pending]
#### Explanation of Calculations:
1. Each coin type is assigned a probability based on its frequency in the box.
2. The expected value \( E(X) \) is calculated as the sum of products of each value and its probability.
3. Variance is derived by taking each value squared, multiplying by its probability, summing all those values, and subtracting the square of the mean.
4. The standard deviation is the square root of the variance, providing insight into the spread of the coin values around the mean.
This illustration explains how to determine the expected value, variance, and standard deviation of a given random variable from a finite probability distribution.
![The webpage appears to be part of an online educational platform from Citrus College. It is related to statistical calculations, specifically focusing on population mean, variance, and standard deviation. Here's a breakdown of the content:
1. **Equations Section**:
- There are placeholders for calculations and several mathematical symbols used to perform statistical operations.
- The equation for variance (σ²) includes terms for summation (x²P(x) and [ΣxP(x)]²), with specific numbers plugged in (15.4 and 3.6).
2. **Calculation Fields**:
- **Population Mean**: There is a dropdown or input field labeled `Select an answer` for entering or selecting the mean.
- **Population Variance**: A similar setup for variance.
- **Population Standard Deviation**: Followed by an input field where the answer should be rounded to two decimal places.
3. **Analytical Questions**:
- Part (c) asks, "What does the population mean tell us about this problem?" with a dropdown to select an answer.
- Part (d) asks, "What does the population standard deviation tell us about this problem?" also with a dropdown to select an answer.
4. **Support Options**:
- There is a "Question Help" button accompanied by a `Message Instructor` option for further assistance.
5. **Submission**:
- A `Submit Question` button is for submitting the answers once completed.
**Navigation and Interface**:
- The left side features tools such as Dashboard, Courses, Calendar, Inbox, and Help.
- The webpage header shows additional open tabs, and the address bar indicates the platform (citruscollege.instructure.com).
This educational interface is designed to guide students through the process of calculating and understanding fundamental statistical measures, facilitating a deeper comprehension of data analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F95e5e52d-4449-4f93-832b-400422add1f0%2F4fb92754-9b35-40de-a9ff-4f984e28d588%2Foqleh3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The webpage appears to be part of an online educational platform from Citrus College. It is related to statistical calculations, specifically focusing on population mean, variance, and standard deviation. Here's a breakdown of the content:
1. **Equations Section**:
- There are placeholders for calculations and several mathematical symbols used to perform statistical operations.
- The equation for variance (σ²) includes terms for summation (x²P(x) and [ΣxP(x)]²), with specific numbers plugged in (15.4 and 3.6).
2. **Calculation Fields**:
- **Population Mean**: There is a dropdown or input field labeled `Select an answer` for entering or selecting the mean.
- **Population Variance**: A similar setup for variance.
- **Population Standard Deviation**: Followed by an input field where the answer should be rounded to two decimal places.
3. **Analytical Questions**:
- Part (c) asks, "What does the population mean tell us about this problem?" with a dropdown to select an answer.
- Part (d) asks, "What does the population standard deviation tell us about this problem?" also with a dropdown to select an answer.
4. **Support Options**:
- There is a "Question Help" button accompanied by a `Message Instructor` option for further assistance.
5. **Submission**:
- A `Submit Question` button is for submitting the answers once completed.
**Navigation and Interface**:
- The left side features tools such as Dashboard, Courses, Calendar, Inbox, and Help.
- The webpage header shows additional open tabs, and the address bar indicates the platform (citruscollege.instructure.com).
This educational interface is designed to guide students through the process of calculating and understanding fundamental statistical measures, facilitating a deeper comprehension of data analysis.
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