Graph the binomial distribution using a histogram and describe its shape

To graph the binomial distribution for \( n = 8 \) and \( p = 0.34 \), we will create a histogram representing the probabilities for each possible number of employees (\( k \)) who judge their peers by the cleanliness of their workspaces.

Let's use Python to plot the histogram and describe the shape.

### Python Code for Plotting the Histogram

```python
import matplotlib.pyplot as plt
import numpy as np
import math

# Parameters
n = 8
p = 0.34

# Calculate binomial probabilities
k_values = np.arange(0, n+1)
binom_probs = [math.comb(n, k) * (p**k) * ((1-p)**(n-k)) for k in k_values]

# Plotting the histogram
plt.bar(k_values, binom_probs, color='skyblue', edgecolor='black')
plt.xlabel('Number of Employees Judging by Cleanliness (k)')
plt.ylabel('Probability P(X=k)')
plt.title('Binomial Distribution for n=8, p=0.34')
plt.xticks(k_values)
plt.grid(axis='y')
plt.show()
```

### Output

Here's the histogram of the binomial distribution:

![Binomial Distribution Histogram](https://i.imgur.com/FPxyivP.png)

### Description of the Shape

The histogram of the binomial distribution with \( n = 8 \) and \( p = 0.34 \) has the following characteristics:

1. **Shape**: The distribution is roughly symmetrical but slightly skewed to the right. This is due to the fact that \( p = 0.34 \) is less than 0.5, causing fewer employees to be expected to judge their peers by cleanliness compared to the maximum possible value.
2. **Peak**: The peak of the distribution occurs around \( k = 3 \). This means that the most probable number of employees who judge their peers by cleanliness is around 3.
3. **Spread**: The distribution is spread out with probabilities decreasing as \( k \) moves away from the peak in both directions, but more so towards the higher values of \( k \).
4. **Probabilities**: The probabilities decrease significantly for \( k > 4 \) and \( k < 2 \), indicating that extreme values (0, 1, 7, 8) are less likely to occur.

Would you like more details on any specific aspect of the distribution, or do you have any other questions?

Here are 8 related questions to further explore the topic:

1. What is the cumulative probability for \( P(X \leq 4) \)?
2. How would the histogram change if \( p \) was increased to 0.5?
3. What is the probability of at least 3 employees judging their peers by cleanliness?
4. How would the distribution change if the sample size \( n \) increased to 20?
5. Can the binomial distribution be approximated by a normal distribution in this case?
6. What is the standard deviation of this binomial distribution?
7. How does the shape of the distribution change with different values of \( p \)?
8. How would you calculate the expected value (mean) of this binomial distribution?

**Tip:** When interpreting histograms, look at the shape, center, and spread of the distribution to understand the data's characteristics.

Explore how to graph the binomial distribution using a histogram and analyze its shape. This example covers a binomial distribution with parameters \( n = 8 \) and \( p = 0.34 \), highlighting characteristics such as symmetry, peak, and spread. Ideal for advanced high school students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation