Project Overview
This project aims to determine the relationship between the type of fund (stocks or bonds) and client satisfaction levels (high, medium, low) in a financial advisory context. The primary objective is to assess if there's a statistically significant association between the type of investment fund and the client satisfaction reported.

Key Question
Is client satisfaction dependent on the type of fund (stocks or bonds) they invest in?

Tools and Technologies
Python Programming Language: Used for performing the statistical analysis.
Libraries Used:
NumPy: For numerical computations.
DataScience: For handling tabular data.
Matplotlib: For creating histograms and other graphical representations.
warnings: To manage warning messages.
Jupyter Notebook: Provides an interactive environment for running the Python code and visualizing data.
Statistical Methods and Analysis
Data Aggregation
The data consists of client satisfaction levels for different types of funds. It's presented in a tabular format showing the count of clients reporting various satisfaction levels, grouped by the type of fund.

Hypothesis Testing
The hypothesis posits that fund type and client satisfaction level are independent of each other. The analysis involves the following steps:

Data Aggregation: Count of each fund type and client satisfaction level.
Probability Calculation: For each fund type-client satisfaction level pair.
Expected Count Calculation: Based on hypothesized probabilities assuming independence.
Chi-Squared Test: Calculation of the sample chi-squared statistic to compare observed and expected frequencies.
P-Value Calculation: To assess the probability of observing the data if the null hypothesis is true.
Results Visualization
Histograms are used to visualize the distribution of the chi-squared statistic under the null hypothesis and to indicate the critical region for the significance level.

Results Interpretation
Chi-Squared Statistic: Determines if the observed distribution of data differs significantly from what would be expected under the hypothesis of independence.
P-Value: Assesses the strength of the evidence against the null hypothesis.
Significance Level (0.05): Used to determine the threshold for rejecting the null hypothesis.
Conclusion
The project concludes by interpreting the p-value and chi-squared statistic in the

context of the hypothesis. If the p-value is less than the significance level (0.05) and the sample chi-squared is greater than the critical value, the hypothesis of independence is rejected. This would suggest that there is a significant association between the type of fund and client satisfaction.

Further Considerations
Sample Size: The size of the data set might affect the power of the test.
Assumptions: The chi-squared test assumes that the counts are independent and sufficiently large.
Additional Analysis: Further analysis could involve more granular data or different subgroups of clients.
Usage and Adaptation
This project serves as a template for statistical analysis in similar scenarios where categorical data needs to be analyzed for independence or association. Users can adapt the code and methodologies to fit their specific datasets and hypotheses.