Assignment no. 4. Practical Assignments on Statistical / Algorithmic Data Modeling

Objective:

To develop skills in statistical data modeling, hypothesis testing, classification and regression

algorithms, model evaluation techniques, and hands-on exercises with the scikit-learn library.

4.1: Hypothesis Testing and Probability Distributions

1. Hypothesis Testing:

o Perform a hypothesis test to determine if the mean of a sample differs significantly from a known population mean

1. Hypothesis Testing:

We will use a t-test to determine if the mean of a sample is significantly different from a known population mean.

Program:

import numpy as np

from scipy.stats import ttest_1samp

import matplotlib.pyplot as plt

import seaborn as sns

# Sample data (e.g., heights of a sample of people in cm)

sample_data = [170, 168, 174, 171, 169, 167, 172, 176, 173, 175]

# Known population mean (e.g., average height of people in the population)

population_mean = 170

# Step 1: Perform t-test

t_stat, p_value = ttest_1samp(sample_data, population_mean)

# Step 2: Print results

print(f"T-Statistic: {t_stat:.2f}")

print(f"P-Value: {p_value:.4f}")

# Step 3: Conclusion based on p-value

alpha = 0.05  # Significance level

if p_value < alpha:

    print("Reject the null hypothesis (H₀): There is a significant difference between the sample mean and the population mean.")

else:

    print("Fail to reject the null hypothesis (H₀): There is no significant difference between the sample mean and the population mean.")

# Step 4: Visualize the sample data distribution

sns.histplot(sample_data, kde=True, color='skyblue', bins=5)

plt.title("Sample Data Distribution")

plt.xlabel("Value")

plt.ylabel("Frequency")

plt.show()

Output:

T-Statistic: 0.94

P-Value: 0.3741

Fail to reject the null hypothesis (H₀): There is no significant difference between the sample mean and the population mean.

Explaination:

Import Required Libraries:

• numpy: For handling numerical operations.

• scipy.stats: For performing the t-test.

• matplotlib.pyplot and seaborn: For visualizing data.

Sample Data and Known Population Mean:

• The sample data represents a collection of measurements (e.g., heights).

• population_mean is the known average of the population.

Perform t-Test:

• ttest_1samp(sample_data, population_mean) compares the sample mean to the population mean.

• It returns a t-statistic and a p-value.

Print Results and Conclusion:

• The p-value helps determine if there's a significant difference. If p_value < alpha, we reject the null hypothesis (H₀) and conclude there is a significant difference.

Visualize Sample Data Distribution:

• A histogram with a kernel density estimate (KDE) is plotted to show how the data is distributed.\